MATHEMATICS, NUMERICS, DERIVATIONS AND OPENFOAM

Transcription

1 MATHEMATICS, NUMERICS, DERIVATIONS AND OPENFOAM The Basics for Numerical Simulations. Tobias Holmann

2 i Dedidcated to the OpenFOAM R communit and especiall to all colleagues and people who support me. The ambition of writing the book is to give an introduction to computational fluid dnamics, show interesting equations and relations that are not given in most of the literature and prepare ou for all the tasks that ou ma focus in our personal career, hopefull with OpenFOAM R. The book is also available as soft-cover. If ou are interested to have on, just ask me for a cop. You can register for free on m homepage to get the latest updates about the book and all of m work or just follow me on Twitter, Facebook, Linkedin, XING or Youtube. Each feedback and all critics are taken into considerations. Please let me know if ou find mistakes or if ou think some special topic is missing.

3 ii Copright Tobias Holmann, fourth edition, 20 September 2017 What is this book about This book collects man aspects of mathematics, numerics and derivations that are used in the field of CFD and OpenFOAM R. Thanksgiving I would thank Dr. Alexander Vakhrushev for all the interesting discussions and the help in the topics of mathematics and programming in OpenFOAM R. Furthermore, I would like to thank Sergei Zappo from the cfd-online platform, Vigneshkumar and Vishwesh Rai Shrimali for the useful remarks and corrections. In addition I thank m sister Michaela Holmann for further improvements and Andrea Jall for the beautiful cover page and all the support in m life. How to cite? The citation format depends on the journal or our personal stle. The following citation is just an example based on the format stle that is used in this document: Tobias Holmann. Mathematics, Numerics, Derivations and OpenFOAMR, Holmann CFD, Leoben, fourth edition, September URL

4 iii Outline This book gives an introduction to the basic mathematics that are used in the field of computational fluid dnamics. After knowing the mathematic aspects, all conservation equations are derived using a finite volume element, dv. It is shown how the mass and momentum equation can be derived. After that all different kinds of energ equations are discussed in all details. Here, the distinguish between the kinetic, the internal, the total energ and the enthalp equation is done. Based on the nature of the equations, the general governing equation is introduced and it is shown how other equations can be derived out of this one. In a separated chapter the definition of the shear-rate tensor τ for Newtonian fluids is given which is followed b a discussion that shows the analog between the Cauch stress tensor σ, the shear-rate tensor τ and the pressure p. The equations are summed up with a one page summar of all all equations. Due to the fact that the flow pattern for engineering applications are mostl turbulent, the Renolds-Averaging methods are explained. After that the incompressible Renolds-Averaged-Navier-Stokes equations are derived and finall the closure problem is discussed. Here, we investigate into the Renolds-Stress equation and the analog to the Cauch stress tensor is shown. During the introduction of the edd-viscosit theor, the equation for the turbulent kinetic energ k, dissipation ɛ are deducted and it is shown how the turbulence modeling is applied. The topic ends with a brief description about the derivation for the compressible equations and its difficulties. At the end of the book, the implementation of the shear-rate tensor τ in OpenFOAM R is given and discussed. Here, the main focus is based on the investigation into the C++ code and the numerical stabiliation. Finall, a more general discussion of the different pressure-momentum coupling algorithms is performed and it is explained how to use the PIMPLE algorithm in a correct wa using OpenFOAM R. The last chapter is related to OpenFOAM R tutorials and some further useful information.

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6 Contents 1 Basic Mathematics Basic Rules of Derivatives Einsteins Summation Convention General Tensor Mathematics The Total Derivative Matrix Algebra, Deviatoric and Hdrostatic Part The Gauss Theorem Derivations of the Governing Equations The Continuit Equation Integral Form of the Conserved Continuit Equation Continuit Equation and the Total Derivative The Conserved Momentum Equation The Proof of the Transformation Integral Form of the Conserved Momentum Equation Non-Conserved Momentum Equation The Conserved Total Energ Equation The Proof of the Vector Transformation Integral Form of the Conserved Total Energ Equation Non-Conserved Total Energ Equation Kinetic Energ and Internal Energ The Conserved Mechanical Energ Equation Integral Form of the Conserved Mechanical Energ Equation Non-Conserved Mechanical Energ Equation The Conserved Thermo Energ Equation Integral Form of the Thermo Energ Equation v

7 vi CONTENTS Non-Conserved Thermo Internal Energ Equation The Conserved Enthalp Equation Integral Form of the Conserved Enthalp Equation Non-conserved Enthalp Equation The Conserved Enthalp Equation onl Thermo The Governing Equations for Engineers The Continuit Equation The Momentum Equation The Enthalp Equation Common Source Terms Summar of the Equations 49 5 The Shear-rate Tensor and the Navier-Stokes Equations Newtonian Fluids The Proof of the Transformation The Term 2 3 µ U Further Simplifications Relation between the Cauch Stress Tensor, Shear-Rate Tensor and Pressure 61 7 The bulk viscosit 63 8 Collection of Different Notations of the Momentum Equations 67 9 Turbulence Modeling Renolds-Averaging Renolds Time-Averaged Equations Incompressible Mass Conservation Equation Compressible Mass Conservation Equation Incompressible Momentum Equation The Incompressible General Conservation Equation The Closure Problem Boussinesq Edd Viscosit Edd Viscosit Approximation Algebraic Models

8 CONTENTS Turbulence Energ Equation Models Incompressible Renolds-Stress Equation The Incompressible Kinetic Energ Equation The Relation between ɛ and L The Equation for the Dissipation Rate ɛ Coupling of the Parameters Turbulence Modeling for Compressible Fluids Calculation of the Shear-Rate Tensor in OpenFOAM R The Inco. Shear-Rate Tensor, divdevreff The Compr. Shear-Rate Tensor, divdevrhoreff Influence of Turbulence Models SIMPLE, PISO and PIMPLE algorithm The SIMPLE algorithm in OpenFOAM R SIMPLEC in OpenFOAM R The PISO algorithm in OpenFOAM R The PIMPLE algorithm in OpenFOAM R The correct usage of the PIMPLE algorithm The test case First considerations Run the case with the PISO algorithm PIMPLE working as PISO PIMPLE working as PISO with large t PIMPLE algorithm modified add outer corrections PIMPLE algorithm further modified add inner corrections PIMPLE algorithm with under-relaxation PIMPLE algorithm speed up PIMPLE conclusion The relaxation methods Field relaxation Matrix relaxation OpenFOAM R tutorials 141

9 2 CONTENTS 14 Appendix The Incompressible Renolds-Stress-Equation

10 Chapter 1 Basic Mathematics In the field of computational fluid dnamics, it is essential to understand the equations and the mathematics. This will be helpful especiall if we are going to implement, reorder or manipulate equations within a software or toolbox. There are a lot of was to represent equations and thus a brief collection of the most essential mathematics are given in this chapter. The beaut of mathematics are also described in Greenshields [2015], Dantig and Rappa [2009], Jasak [1996] and Moukalled et al. [2015]. In the field of numerical simulations we are dealing with tensors T n of rank n. A tensor stands for an kind of field like a scalar, a vector or the classical known tensor that represents a matrix normall a 3 b 3 matrix and is of rank two. To keep things clear we use the following definition: Zero rank tensor T 0 := scalar a First rank tensor T 1 := vector a Second rank tensor T 2 := tensor T matrix of 3x3 Third rank tensor T 3 := tensor T ijk A tensor that is of higher order than rank ero is alwas given in bold smbol. Tensors higher than second order are onl needed during the derivation of the Renolds-Stress equation. 3

11 4 Basic Mathematics 1.1 Basic Rules of Derivatives The governing conservation equations in fluid dnamics are partial differential equations. Thats wh we briefl summarie the rules that are needed when we deal with this kind of equations. Considering the sum of two quantities φ and χ that are derived respectivel to τ, we can split the derivative: φ + χ τ = φ τ + χ τ. 1.1 If we have to derive the product of the two quantities, we need to use the product rule to split the term. In other words, we have to keep one quantit constant whereas we derive the other one: φχ τ = χφ τ + φχ τ. 1.2 A constant quantit C can be taken inside or outside of a derivative: C φχ τ = C φχ τ Einsteins Summation Convention For vector and tensor equations there are several options of notations. The longest but clearest notation is the Cartesian one. This notation can be abbreviated using the Einsteins summation convention. In general, we are using this convention mostl for vector and tensor quantities. Assuming the sum of derivatives of the arbitrar variable φ i like the momentum in x, and direction, the Cartesian form is written as: φ x x + φ + φ. 1.4 To simplif this equation, we can use the Einsteins summation convention. Commonl we neglect the summation sign to keep things clear and short: i φ i x i = φ i x i i = x,,. 1.5 A more complex example that demonstrates the advantage of the Einsteins summation convention would be the convective term of the momentum equation till now we do not need to know what this equation means and hence, we do not think

12 1.3. GENERAL TENSOR MATHEMATICS 5 about the meaning. Due to the fact that the momentum is a vector quantit, we get three equations: u x u x x + u u x + u u x, 1.6 u x u x + u u + u u, 1.7 u x u x + u u + u u. 1.8 B using the Einsteins convention we can simplif the three equations into one: i u i u j x i = u iu j x i i = x,, ; j = x,,. 1.9 The Einsteins summation convention is widel used in literatures. Hence, we should keep in mind what it stand for and how we have to appl it. 1.3 General Tensor Mathematics A common and eas wa to deal with equations is using the vector notation instead of the Einsteins summation convention. The vector notation require knowledge of special mathematics. We will familiarie different operations that act on scalars, vectors and tensors. For that purpose we introduce a scalar φ, two vectors a and b and a tensor T: a = a x a = a 1 a 2, b = b x b = b 1 b 2, a a 3 b b 3 T xx T x T x T 11 T 12 T 13 T = T x T T = T 21 T 22 T 23. T x T T T 31 T 32 T 33 Depending on the operation we are investigating, we use either the numeric indices 1, 2, 3 or the space components x,,. Furthermore, we need the unit vectors

13 6 Basic Mathematics e i and the identit matrix I: e 1 = e x = 0, e 2 = e = 1, e 3 = e = 0, I = Simple Operations The multiplication of a scalar φ b a vector b results in a vector and is commutative and associative. This is also valid for the multiplication of a scalar φ and a tensor T: φb x φt xx φt x φt x φb = φb, φt = φt x φt φt φb φt x φt φt The Inner Product The inner product of two vectors a and b produces a scalar φ and is commutative. This operation is indicated b the dot sign : 3 φ = a b = a T b = a i b i The inner product of a vector a and a tensor T produces a vector b and is non-commutative if the tensor is non-smmetric: T a 1 + T 12 a 2 + T 13 a 3 b = T a = T ij a j e i = T 21 a 1 + T 22 a 2 + T 23 a i=1 j=1 T 31 a 1 + T 32 a 2 + T 33 a 3 a T 11 + a 2 T 21 + a 3 T 31 b = a T = T T a = a j T ji e i = a 1 T 12 + a 2 T 22 + a 3 T 32, 1.13 i=1 j=1 a 1 T 13 + a 2 T 23 + a 3 T 33 A smmetric tensor is given, if T ij = T ji and hence, a T = T a. i=1 The Double Inner Product The double inner product of two tensors T and S results in a scalar φ and is commutative. It will be indicated b the colon : sign:

14 1.3. GENERAL TENSOR MATHEMATICS 7 φ = T : S = 3 3 T ij S ij = T 11 S 11 + T 12 S 12 + T 13 S 13 + T 21 S 21 i=1 j=1 + T 22 S 22 + T 23 S 23 + T 31 S 31 + T 32 S 32 + T 33 S The Outer Product The outer product of two vectors a and b, also known as dadic product, results in a tensor, is non-commutative and is expressed b the dadic sign : a x b x a x b a x b T = a b = ab T = a b x a b a b a b x a b a b In most of the literatures the dadic sign is neglected for brevit as shown below: ab Keep in mind, that both variants are used in literature whereas the last one is more common but the first one is more clear. In this book we use the definition of equation 1.15, to be more consistent with the mathematics. Differential Operators In vector notation, the spatial derivatives of a variable scalar, vector or tensor is made using the Nabla operator. It contains the three space derivatives of x, and in a Cartesian coordinate sstem: Gradient Operator = x = x 1 x 2 x 3. The gradient of a scalar φ results in a vector a: grad φ = φ = φ x φ φ. 1.17

15 8 Basic Mathematics The gradient of a vector b results in a tensor T: grad b = b = x b x b x b x x b b b x b b b We see that this operation is actuall the outer product of the Nabla operator special vector and an arbitrar vector b. Hence, it is commonl written as: b In this book we use the first notation with the dadic sign to be more consistent within the mathematics. Note: The gradient operation increase the rank of the tensor b one and hence, we can appl it to an tensor field. Divergence Operator The divergence of a vector b results in a scalar φ and is expressed b the combination of the Nabla operator and the dot sign, : div b = b = 3 i=1 x i b i = b 1 x 1 + b 2 x 2 + b 3 x The divergence of a tensor T results in a vector b: T x 1 + T 21 x 2 + T 31 x 3 div T = T = T ji e i = T 12 x x i=1 j=1 j 1 + T 22 x 2 + T 32 x T 13 x 1 + T 23 x 2 + T 33 x 3 Note: The divergence operation decrease the rank of the tensor b one. Hence, it does not make sense to appl this operator on a scalar. The Product Rule within the Divergence Operator If we have a product within a divergence term, we can split the term using the product rule. Based on the tensor ranks inside the divergence, we have to appl different rules, which are given now.

16 1.3. GENERAL TENSOR MATHEMATICS 9 The divergence of the product of a vector a and a scalar φ can be split as follow and results in a scalar: aφ = a φ + φ a Eqn simple multiplication The divergence of the outer product dadic product of two vectors a and b can be split as follow and results in a vector: a b = a } {{ b } + b }{{ a} Eqn Eqn The divergence of the inner product of a tensor T and a vector b can be split as follow and results in a scalar: T b = T } : {{ b } + b } {{ T} Eqn Eqn If ou think that the product rule for the inner product of two vectors is missing, just think about the result of the inner product of two vectors and how this tensor will change rank if we appl the divergence operator. Hopefull ou will figure out, that the divergence of a scalar does not make sense The Total Derivative The definition of the total derivative of an arbitrar quantit φ in the field of fluid dnamics is defined as: Dφ Dt = φ t + U φ inner product, 1.25 where U represents the velocit vector. The last term in equation 1.25 denotes the inner product. Depending on the quantit φ scalar, vector, tensor, and so on, we have to use the correct mathematical expressions for the second term on the right hand side. Example given: If φ is a scalar, we have to use equation 1.11, If φ is a vector, we have to use equation 1.13.

17 10 Basic Mathematics Short Outline for the Total Derivative The total derivative is used to represent non-conserved equations. In other words, each conserved equation can be changed into a non-conserved equation using the continuit equation. In literature people start to derive equations using the total derivative and using the continuit equation to extend the non-conservative equation to the conserved one. The better wa would be to derive first the conserved equation and then using the continuit equation to get the non-conserved form. Wh? It is easier to understand. The difference between both equations is the frame of reference. In the conserved representation, we have the Euler expression, for non-conserved equations it is the Lagrange expression. If ou have literature that start with the non-conservation equations, this would help to understand the following extension at the moment we do not need to understand this equations, it is just an example: Incompressible: Dφ Dt = φ t + U φ non conserved + φ U continuit = Compressible: ρ Dφ [ ] φ Dt = ρ t + U φ non conserved ρ + φ t + ρu continuit = The reason we multipl the continuit equation second term on the right hand side b the quantit φ comes from the product rule, that is applied to the convective term. After the momentum equation is derived and the conservative form is transformed into the non-conserved one, this will get clear Matrix Algebra, Deviatoric and Hdrostatic Part In the field of numerical simulations we are dealing with quantities that are represented b matrices like the stress tensor. Thats wh we need to introduce some basics here. Each matrix A can be split into a deviatoric A dev and hdrostatic A hd part: A = A hd + A dev. 1.28

18 1.3. GENERAL TENSOR MATHEMATICS 11 The hdrostatic part can be expressed as scalar or matrix and is defined b using the trace operator. If we want to get the scalar, we use the following definition: A hd = 1 3 tra = 1 3 n a ii i=1 The operator tr denotes the trace operator and is applied on the matrix. This operator is simpl the sum of the diagonal elements. The matrix notation of the hdrostatic part given b: A hd = A hd I = 1 3 trai = 1 3 n a ii I i=1 The deviatoric part A dev is given as: A dev = A A hd = A 1 trai Note: The deviatoric part of a matrix is traceless. Hence, tra dev = 0; The trace operator is ero not the diagonal elements The Gauss Theorem To transform an equation from the differential form to the integral one or vice versa, it is necessar to know the Gauss theorem. This theorem allows us to establish a relation between the fluxes through the surface of an arbitrar volume element and the divergence operator on the volume element: a nds = adv In equation 1.32, n represents the surface normal vector pointing outwards, ds the integration with respect to the surface and dv the integration with respect to the volume. Note: The small dot denotes the inner product of two vectors In the following book, we use the small dot in all integrals to sign that we calculate the inner product of a vector a and the surface normal vector n. Keep in mind that the small dot expresses exact the same as the bullet.

19 12 Basic Mathematics

20 Chapter 2 Derivations of the Governing Equations The following chapter demonstrates how to derive the continuit, momentum, total energ, mechanical kinetic energ, thermo internal energ and enthalp equation using a small volume element dv. The equations are derived using the Cartesian coordinate sstem. A complete summar of all equations is given on page 49. The structure of this chapter is mainl as follows: Express the phenomena that act on the volume element using finite differences, Transform the finite difference equation to a partial differential equation, Manipulate the equation to get the common form, Transform the Cartesian notation into the vector notation, Proof that the vector notation results in the Cartesian notation, Transform the equation into the integral and non-conserved form. The main references that are used within this chapter are Greenshields [2015], Dantig and Rappa [2009], Jasak [1996], Feriger and Perić [2008], Bird et al. [1960], Versteeg and Malalasekera [1995], Schware [2013] and Moukalled et al. [2015]. 13

21 14 Derivations of the Governing Equations 2.1 The Continuit Equation In the following section we are going to derive the continuit equation. This equation is simpl a mass balance of an arbitrar volume element. Consider the mass flow through a small control volume element dv, using the constrain that mass is not transformed into energ or vice versa, a mass balance has to be fulfilled for the volume element. That means, that the mass flow that enters or leaves the volume element through its surfaces has to be equal. Furthermore, we have to take the rate of mass accumulation into account: [ ] [ ] [ ] rate of mass rate of mass rate of mass =. 2.1 accumulation entering the volume leaving the volume To make things clearer, we anale figure 2.1. It is obvious that the mass is transported through the surface b the velocit. This transport phenomenon is called convection or sometimes named advection normall if we are talking about the fluid itself, we refer to convection; however, if a species or an other quantit is transported due to convection, then we talk about advection and happens for all three space directions x u x, u and u. Additionall a mass change inside the element can occur due to compression or expansion phenomena; that means the densit will change. Having a closer look on figure 2.1, we can see that it shows the velocit vectors normal to the faces. The rate of mass that enters or leaves the volume element Mass inside x : ρu x x u x x dv x u x x+ x Mass outside x : ρu x x+ x Mass inside : ρu x Mass outside : ρu + x Mass inside : ρu x Mass outside : ρu + x x Figure 2.1: Mass balance in a small volume element dv.

22 2.1. THE CONTINUITY EQUATION 15 through the surface is called mass flux and is simpl the densit times the velocit with respect to the area of the face. For the derivation of the mass conservation equation we have to build the balance of the fluxes at the surfaces of the volume element. In other words, everthing that is going inside has to go out if we assume that there is no mass accumulation inside the volume incompressible. The single terms that describe the fluxes at the surfaces are given on the right side of figure 2.1. If we have a compressible fluid, the rate of change for the densit ρ is related to the volume, mass per unit volume, and will onl change with respect to time. Therefore, we can write the rate of change of the densit as: ρ t. 2.2 Rewriting equation 2.1 b using the mathematical expressions of figure 2.1 and equation 2.2, it follows: ρ t x = ρu x x ρu x x+ x + ρu ρu + x + ρu ρu + x. 2.3 Dividing the equation b the volume V = x, we get: ρ t = ρu x x ρu x x+ x x + ρu ρu + + ρu ρu Introducing the assumption of an infinitesimal small volume element which means that we decrease the distance between the corners of the volume element and therefore goes to ero: x lim x 0 x = x, 2.5

23 16 Derivations of the Governing Equations and also an infinitesimal small time range: t lim t 0 t = t, 2.6 we can transform the finite difference equation to a partial differential equation. For that, we have to appl equation 2.5 and 2.6 to 2.4. It follows: ρu x x ρu x x+ x x ρu ρu + = ρu x x = ρu x ρu x, 2.7 ρu, 2.8 ρu ρu x + = ρu ρu, 2.9 ρ t ρ t, 2.10 and the general mass conservation continuit equation is given b: ρ t = x ρu x + ρu + ρu If we use the Nabla-Operator and the velocit vector U, the equation can be rewritten in vector notation: ρ t = ρu However, if we focus on incompressible fluids, we could assume that the densit is constant and therefore the quantit ρ can be taken out of the derivatives and we are allowed to divide b ρ. It is obvious that the time derivative will vanish because of the fact that the densit is a constant and will not change with respect to time. One ma also tr to explain it in the following wa: if we assume constant densit, there is no expansion or compression phenomena and therefore the time derivative can be canceled to ero. Hence, onl the mass flux that enters and/or leaves the volume element at its surface has to be taken into account. Remark: In man cases incompressibilit means that there is no expansion and/or compression phenomena. However, the fluid densit can still be temperature depended. In such a case we have to be careful which mass conservation

24 2.1. THE CONTINUITY EQUATION 17 equation we are using incompressible or compressible. In general, if the densit is not a constant value, we are not allowed to use the simplified mass conservation equation 2.13 due to the fact that it is not possible anmore to push the densit out of the derivative. If the densit change is reall small, we are allowed to use the simplified mass conservation equation with limitations. The reason for that is based on numerics and the interaction with the momentum conservation equation. For the incompressible case, the densit for the fluid is constant and thus we can simplif the mass conservation equation to: U = For the simple mass conservation equation it is obvious that the vector notation results in the Cartesian form. Thats wh the transformation is not demonstrated here. If ou want to check it ourself, ou just need to use equation Integral Form of the Conserved Continuit Equation For the completeness, the integral form of the mass conservation equation will be given now. Using the Gauss theorem 1.32, we can transform the divergence term that acts on the volume to a surface integral. The accumulation of densit in the element is a simple volume integral. Furthermore, the volume element itself is not changing with respect to time fixed finite volume - the discrete volumes are not changing during the simulation; static meshes. Thus, we end up with: Compressible: t ρdv = ρu nds Incompressible: U nds = The surface integral means nothing more than taking the balance of the fluxes on the surface of volume element; what is going in and out. Depending on the shape of the volume, we have to evaluate more or less faces. The integral form of the continuit equation leads to the so called finite volume method FVM. This method is conservative and we can appl this method to each arbitrar volume hexaeder, tetraeder, prisms, wedges and so on which makes this method popular.

25 18 Derivations of the Governing Equations Continuit Equation and the Total Derivative Using the total derivative formulation 1.25, we are able to rewrite the continuit equation 2.12 b appling the product rule 1.22 to the divergence term: ρu = U ρ + ρ U Substituting this expression into equation 2.12, we get: ρ t Finall, we put all terms to the LHS: The result is: = U ρ ρ U ρ t + U ρ +ρ U = Total derivative Dρ + ρ U = Dt This equation is not common but could be found in Anderson [1995]. OpenFOAM R In OpenFOAM R we are using this equation integral one to calculate the fluxes at the faces of each cell. The flux field is named phi in and is created b the call of one of the two header files in each solver: createphi.h compressiblecreatephi.h Due to the fact that we store the densit and the velocit at the cell center, we need to interpolate the values to the face centers. This is done b calling the function interpolaterho*u. This will simpl calculate the product of the densit and the velocit vector at the cell center and interpolate it to the face center b including the neighbor cell information with respect to the face we are going to evaluate. To get the fluxes, the interpolated values are then multiplied b the surface normal vector area using the inner product of two vectors denoted b the ampersand sign & in OpenFOAM R.

26 2.2. THE CONSERVED MOMENTUM EQUATION The Conserved Momentum Equation For the derivation of the conserved momentum equation, we are going to use the volume element dv again. The main difference in the momentum equation compared to the mass conservation equation is, that we have to consider more phenomena that can transport and change the momentum inside the volume element and that this quantit is not a scalar; it is a vector velocit in x, and direction. Generall we are allowed to sa that the momentum can be transported and changed b the following aspects: rate of rate of rate of sum of forces momentum = momentum entering the momentum leaving the + that act on accumulation the volume volume volume Figure 2.2 shows the volume element like in 2.1 but now showing another transport phenomenon that acts onl on the surfaces. The phenomenon transports the momentum based on molecular effects. This molecular transport acts normal and tangential to the surface and is an outcome or propert of the vector quantit. Other phenomena that change the momentum are given as a sum of forces acting on the volume element dv. For example we could have the gravitational acceleration and the pressure force. τ x + into face x : τ xx x x out of face x : τ xx x+ x τ x + into face : τ x x τ xx x τ xx x+ x out of face : τ x + x τ x into face : τ x x τ x x out of face : τ x + x Figure 2.2: Molecular transport of the momentum in x-direction in an arbitrar small volume element dv.

27 20 Derivations of the Governing Equations On the right side of figure 2.2, the terms that transport the x-component of the momentum through the surfaces b the molecular transport effect are given. Convection of the Momentum in x-direction The x-component of the momentum is transported b convection into the volume element through all six faces that is also an outcome related to the vector quantit. Therefore, the convection of the momentum can be derived similarl to the convective transport of the mass. But now we have to take care about the vector quantit. Thus, the momentum in x-direction enters the volume at the face x and leaves the volume through the face x+ x ; identical to the continuit equation. But now it is also possible that the x-component of the momentum is transported through the faces in and direction. Hence, we can write the transport of the momentum due to convection; it is simpl the velocit in x-direction multiplied b the flux through the face we are looking at Newtons second law: into face x : ρu x u x x, out of face x+ x : ρu x u x x+ x, into face : ρu u x, out of face + : ρu u x +, into face : ρu u x, out of face + : ρu u x +. After combining the terms and using the face areas, we get: ρux u x x ρu x u x x+ x + ρu u x ρu u x + x + ρu u x ρu u x + x. Molecular Transport of the Momentum in x-direction Additionall, the x-component of the momentum is transported due to the molecular phenomenon as demonstrated in figure 2.2. The effect is based on velocit differences velocit gradients. As we can see in figure 2.2, we have different kind of terms: the normal component τ xx and the tangential components τ x, τ x. Therefore, the molecular transport of the x-momentum through the surfaces can

28 2.2. THE CONSERVED MOMENTUM EQUATION 21 be written as: τxx x τ xx x+ x + τ x τ x + x + τ x τ x + x. These terms represent additional fluxes of momentum through the surface. We consider these fluxes as stresses. τ xx denotes the stress perpendicular to the direction we are looking at here face x and face x+ x and τ x, τ x denote the x-directed tangential stresses which act on the faces with respect to the indices. All these stresses are known as shear stresses due to the fact that the are generated with respect to velocit gradients that introduce shearing. Additional Forces that Influence the Momentum In most problems, the onl important forces that influence the momentum are the pressure and gravit force. The pressure acts on the surface whereat the gravitational force acts on the volume of the element. Hence, we are able to derive the change of the x-momentum based on the pressure and gravitational force: p x p x+ x + ρgx x. Conserved Momentum Equation After we have all terms, we can reconstruct equation 2.20 with the mathematic expressions. Of course the accumulation of the momentum inside an arbitrar volume element is given b: t ρu x x.

29 22 Derivations of the Governing Equations Thus, for the momentum in x-direction we can write: t ρu x x = ρu x u x x ρu x u x x+ x + ρu u x ρu u x + x + ρu u x ρu u x + x + τ xx x τ xx x+ x + τ x τ x + x + τ x τ x + x + p x p x+ x +ρg x x Now, b dividing the whole equation b the volume dv, it follows: t ρu x = ρu xu x x ρu x u x x+ x x + ρu u x ρu u x + + ρu u x ρu u x + + τ xx x τ xx x+ x x + τ x τ x + + τ x τ x + + p x p x+ x x + ρg x Finall we use the assumption of an infinitesimal small volume element 2.5 and time range 2.6 to rewrite the x-component of the momentum equation above. The other two space directions can be derived in the same wa and is not shown in details. The x-component of the momentum is then written as: t ρu x = x ρu xu x + ρu u x + ρu u x x τ xx + τ x + τ x p x + ρg x For the -component of the momentum, we get: t ρu = x ρu xu + ρu u + ρu u x τ x + τ + τ p + ρg, 2.24

30 2.2. THE CONSERVED MOMENTUM EQUATION 23 and for the -component we achieve: t ρu = x ρu xu + ρu u + ρu u x τ x + τ + τ p + ρg Introducing the gravitational acceleration vector g, the gradient of the pressure p and the shear-rate tensor τ that are defined as p = p x p p τ xx τ x τ x, τ = τ x τ τ, g = τ x τ τ we are able to write the momentum equation in vector form. g x g g, Note, that the negative sign of the shear-rate tensor will change, if we introduce the definition of the shear-rate tensor τ later on. ρu = ρu U τ p + ρg 2.26 t The Proof of the Transformation The following section will proof that equation 2.26 results in 2.23, 2.24 and For clearance, we will focus on each term separatel. Starting with the first term, the time derivative, we get: t ρu = t ρu x t ρu t ρu =! t ρu x t ρu t ρu of x momentum of momentum of momentum As we see, the time derivative term results in the same three terms that we have in the Cartesian formulation. The second term embrace the transport of momentum due to convection b the flux ρu. To evaluate the term, we need the mathematics 1.15 and 1.21: ρu U = ρ u x u u u x u u u x u x u x u u x u = ρ u u x u u u u u u x u u u u

31 24 Derivations of the Governing Equations ρu x u x ρu x u ρu x u x ρu x u x ρu x u ρu x u = ρu u x ρu u ρu u = ρu u x ρu u ρu u ρu u x ρu u ρu u ρu u x ρu u ρu u x ρu xu x + ρu u x + ρu u x x ρu xu x + ρu u x + ρu u x = x ρu xu + ρu u + ρu u =! x ρu xu + ρu u + ρu u x ρu xu + ρu u + ρu u x ρu xu + ρu u + ρu u Again, it can be seen that the terms are equal and we end up with the same set of equations. Now we investigate into the third term that describes shearing due to the gradients of the velocities. Using the convention 1.21 for the shear-rate tensor τ, we get: τ xx τ x τ x τ = τ x τ τ = τ x τ τ x τ xx + τ x + τ x x τ xx + τ x + τ x of x momentum x τ x + τ + τ =! x τ x + τ + τ of momentum x τ x + τ + τ x τ x + τ + τ of momentum It was alread clear, that we end up with the same terms. At last the pressure and gravitational acceleration term is analed. For the pressure term we need the definition of equation 1.17 and for the gravitational term equation It follows: p = ρg = ρ x g x g g p = p x p p =! p x p p of x momentum of momentum of momentum ρg x ρg x of x momentum = ρg =! ρg of momentum ρg ρg of momentum As we proofed now, the vector form ends up in the Cartesian form. If we want to solve this equation now, it is necessar to know the shear-rate tensor τ. We are going to investigate into that quantit in chapter 5. Further.,..

32 2.2. THE CONSERVED MOMENTUM EQUATION 25 representations of the momentum equation can be found in chapter 8. The implementation of the momentum equation in OpenFOAM R will be discussed in chapter 10; especiall the treatment of the diffusion term. Keep in mind that equation 2.26 includes onl the gravitational acceleration and pressure force. If there are further phenomena forces influencing the momentum equation, these terms have to be taken into account Integral Form of the Conserved Momentum Equation The integral form of the momentum equation 2.26 can be obtained b using the Gauss theorem It follows: t ρudv = ρu U nds τ nds pi nds + ρgdv Non-Conserved Momentum Equation We can manipulate the conserved momentum equation with the continuit equation 2.12 to get a non-conservative form. For that, we consider equation 2.26 first and split the time and convection term b using the product rule. The time derivative becomes: t ρu = ρ t U + U t ρ, 2.29 and the convection term can be rewritten using equation It follows: ρu U = ρu }{{ U } +U ρu gradient divergence inner product Replacing these terms into equation 2.26 and put the convection terms to the LHS, we end up with: ρ t U + U ρ + ρu U + U ρu = t

33 26 Derivations of the Governing Equations After analing the equation, we see that we can take out ρ and U: [ ] ρ t U + U U + U [ ] t ρ + ρu = continuit It is clear that the second term is ero due to the continuit equation. Appling the definition of the total derivative 1.25, we can write the non-conservative form of the momentum equation as: ρ DU Dt = τ p + ρg 2.33 Remark: As alread mentioned before, the negative sign of the first term on the RHS will vanish after we introduced the definition of the shear-rate components τ ii. 2.3 The Conserved Total Energ Equation This section will show the derivation of the total energ equation. The total energ includes the internal thermal and kinetic mechanical energ. In general, the change of the total energ can be described in an arbitrar volume element dv b: rate of internal and kinetic rate of internal rate of internal = and kinetic energ and kinetic energ energ accumulation entering the volume net rate of net rate of + heat addition b work done b conduction sstem on surroundings leaving the volume net rate of + additional. heat sources 2.34 This is the first law of thermodnamics written for an open and unstead state sstem with the extension of additional heat sources which was also stated b Bird et al. [1960]. The statement is not complete because no transport of energ can be done due to nuclear, radiative and electromagnetic phenomena but as we alread mentioned, we assume that energ cannot be transfered into mass and vice versa. In the equation above, internal, kinetic and work energ are included and therefore unstead behavior is allowed. The kinetic energ per unit mass is given b

34 2.3. THE CONSERVED TOTAL ENERGY EQUATION ρ U 2 where U denotes the magnitude of the local velocit. The internal energ e per unit mass can be interpreted as the energ associated with the random translation and internal motion of molecules plus the energ of interaction between them. Therefore, the internal energ is temperature and densit depended. The Accumulation of Total Energ during Time Now we write the above equation explicit for a finite volume element dv. The accumulation in time is clear like in the other equations before: x t ρe ρ U The Convection of Total Energ To get the net rate of total energ that enters and leaves the volume element based on the convection phenomenon, we simpl have to multipl the internal and kinetic energ b the velocit respectivel to the face it acts on compare figure 2.1: [u x ρe + 12 ρ U 2 x u x ρe + 12 ] ρ U 2 x+ x + x [u ρe + 12 ρ U 2 u ρe + 12 ] ρ U x [u ρe + 12 ρ U 2 u ρe + 12 ] ρ U If we take out the densit, we clearl see the mass flux; e.g. u x ρ. The Change of Total Energ due to Conduction The next term that influences the energ is based on the conduction phenomenon. For this we introduce the heat flux vector q, that is used later on: [q x x q x x+ x ] + x [q q + ] + x [q q + ] The quantities q x, q and q are the single components of the heat flux vector q.

35 28 Derivations of the Governing Equations The Change of Total Energ due to Work against its Surroundings The work done b the fluid against its surroundings can be split into two parts: The work against the volume forces like gravit, The work against the surface forces like pressure or viscous forces. Some recall: Work = Force x Distance in the direction of the force, rate of doing work = Force x Velocit in the direction of the force. Hence, the rate of doing work against the components of the gravitational acceleration can be written as: ρ x u x g x + u g + u g, 2.38 and the rate of doing work against the pressure p static pressure at the faces of the volume element is: [ pu x x + pu x x+ x ] + x [ pu + pu + ] + x [ pu + pu + ] In a similar wa, the rate of doing work against the viscous forces is: [ τ xx u x + τ x u + τ x u x + τ xx u x + τ x u + τ x u x+ x ] + x [ τ x u x + τ u + τ u + τ x u x + τ u + τ u + ] + x [ τ x u x + τ u + τ u + τ x u x + τ u + τ u + ] Additional Heat Source Additional heat sources or sinks can be taken into account b simpl defining a source term: Q S = x ρs Here Q S denotes the heat source term. The subscript S stands for Source or Sink.

36 2.3. THE CONSERVED TOTAL ENERGY EQUATION 29 Now we have all terms and are able to rewrite equation 2.22: x t ρe ρ U 2 = [u x ρe + 12 ρ U 2 x u x ρe + 12 ] ρ U 2 x+ x + x [u ρe + 12 ρ U 2 u ρe + 12 ] ρ U x [u ρe + 12 ρ U 2 u ρe + 12 ] ρ U [q x x q x x+ x ] + x [q q + ] + x [q q + ] +ρ x u x g x + u g + u g [ pu x x + pu x x+ x ] x [ pu + pu + ] x [ pu + pu + ] [ τ xx u x + τ x u + τ x u x + τ xx u x + τ x u + τ x u x+ x ] x [ τ x u x + τ u + τ u + τ x u x + τ u + τ u + ] x [ τ x u x + τ u + τ u + τ x u x + τ u + τ u + ] + x ρs As before, we divide everthing b the volume dv and use the assumption 2.5. Please keep in mind, that we have derivatives of φ x φ x+ x that end up with a negative sign and φ x+ x φ x that end up with a positive sign. Hence, we get: t ρe ρ U 2 = [ u x ρe + 1 ] x 2 ρ U 2 [ u ρe + 1 ] 2 ρ U 2 [ u ρe + 1 ] 2 ρ U 2 x [q x] [q ] [q ] + ρ u x g x + u g + u g x [pu x] [pu ] [pu ] + x [ τ xxu x + τ x u + τ x u ] + [ τ xu x + τ u + τ u ] + [ τ xu x + τ u + τ u ] +ρs After taking out the minus signs and sort the equation, we get the conserved total

37 30 Derivations of the Governing Equations energ equation as: t ρe + 1 { [ 2 ρ U 2 = u x ρe + 1 ] x 2 ρ U 2 + [ u ρe + 1 ] 2 ρ U 2 + [ u ρe + 1 ]} { 2 ρ U 2 x q x + q + } q + ρ u x g x + u g + u g { x pu x + pu + } { pu x τ xxu x + τ x u + τ x u + τ xu x + τ u + τ u + } τ xu x + τ u + τ u + ρs To get a more visible and readable equation, we use the vector notation. The total energ equation then can be written in the following form and the terms can be described more precisel: t ρe ρ U 2 = ρue U 2 convection pu pressure [τ U] viscous forces q conduction + ρs heat source + ρ U g gravit All terms on the RHS denote the inner product of two vectors Hence, for a implementation into a software toolbox we need to use equation 1.11 and Remark: Till now no word is said about the potential energ. This will not be discussed here. If ou need information about that, ou will find all necessar information in Bird et al. [1960] p The reason wh we do not mention this, is the fact that in most of the engineering cases, the potential energ is not of interest or is in a neglect-able range The Proof of the Vector Transformation The next sites will convert the vector form of the total energ equation back into the Cartesian coordinate sstem. Hence, equation 2.45 will be used and transformed into This is done step b step but not for each term. The terms of interest are the gravit and viscous force term. To manipulate the gravit term we need the mathematic law of equation For the viscous force term we need equation 1.12 and 1.21.

38 2.3. THE CONSERVED TOTAL ENERGY EQUATION 31 It follows that the gravit term can be changed as: ρu g = ρ u x u g x g =! ρ u x g x + u g + u g u g The viscous force term can be changed as follows: τ xx τ x τ x u x [τ U] = τ x τ τ u τ x τ τ u τ xx u x + τ x u + τ x u x τ xx u x + τ x u + τ x u = τ x u x + τ u + τ u = τ x u x + τ u + τ u τ x u x + τ u + τ u τ x u x + τ u + τ u {! = x τ xxu x + τ x u + τ x u + τ xu x + τ u + τ u + } τ xu x + τ u + τ u As we ma alread might had the feeling, the terms are equal of both notations. The other terms that are not discussed above are similar to the momentum equation and it is eas to demonstrate that each term of the vector notation represents the corresponding terms in the Cartesian equation. Due to that, we will not demonstrate it here again Integral Form of the Conserved Total Energ Equation To obtain the integral form of the total energ equation 2.54, we use the Gauss theorem. It follows: t ρe U 2 dv = + ρu gdv ρue U 2 nds pu nds τ U nds + q nds ρsdv Non-Conserved Total Energ Equation As before, the mass conservation equation can be used to manipulate the conserved total energ equation. The transformation lead to the non-conservative form. For

39 32 Derivations of the Governing Equations that, we first need to break the time and convective term of equation 2.45 b using the product rule. Hence, the time derivative can be rewritten as: t ρe+ 1 2 ρ U 2 = [ ρe + 1 ] t 2 U 2 = ρ t e+ 1 2 U 2 +e+ 1 2 U 2 t ρ and the convective term will be transformed to: ρue U 2 = ρu e U 2 +e U 2 ρu } gradient {{ } divergence inner product The next step is replacing the above terms into equation 2.45 and put the terms that are related to the convective part to the LHS: ρ t e U 2 + e U 2 t ρ + ρu e U 2 + e U 2 ρu =... After taking out the densit and the term e U 2, we get: [ ρ t e U 2 + U e + 1 ] 2 U e U 2 [ ] t ρ + ρu =... continuit 2.52 We can observe that the second term on the LHS is equal to ero due to the continuit equation. Based on that, we can simplif the equation and end up with: [ ρ t e U 2 + U e + 1 ] 2 U 2 = B using the definition of the total derivative 1.25, we finall are able to rewrite the equation in the non-conservative form: ρ D Dt e U 2 = q + ρu g pu [τ U] + ρs Kinetic Energ and Internal Energ The total energ equation is the sum of the kinetic energ and internal energ. After we have either the kinetic or internal energ equation, we can simpl subtract that equation from the total energ equation to get the other one. In our case we will

40 2.4. THE CONSERVED MECHANICAL ENERGY EQUATION 33 derive the mechanical kinetic energ equation first and get the internal energ equation b subtracting the kinetic energ from the total energ equation. The answer wh we derive the mechanical kinetic energ equation instead of the internal energ equation is simple. The derivation of the kinetic energ equation can be done simpl b multipling the momentum equation b the velocit again. 2.4 The Conserved Mechanical Energ Equation As mentioned before, we will derive the mechanical kinetic energ equation using the momentum equation. To be consistent with the derivations before, we will split the derivation into two parts. The first part will use the finite volume element dv to derive the first part of the conserved kinetic energ equation without explaining the meaning of the source terms. After that we use the source terms of the momentum equation to get the source terms for the mechanical kinetic energ equation. At that stage it is easier to anale the meaning of the different terms. In general we can define the change of kinetic energ in an arbitrar volume element dv as: [ ] [ ] [ ] rate of kinetic rate of kinetic energ rate of kinetic energ = energ accumulation entering the volume leaving the volume [ ] + sum of additional source terms Using the definition of the kinetic energ per unit mass divided b ρ: 2.55 e kin = 1 2 U 2, 2.56 we can simpl derive the accumulation of the kinetic energ as: x t ρe kin = x t ρ1 2 U The Convection of Kinetic Energ The kinetic energ that enters or leaves the volume element is transported due to fluxes and can be derived analogous to the convective transport of the total energ

41 34 Derivations of the Governing Equations or momentum: into face x : ρu x 1 2 U 2 x, out of face x+ x : ρu x 1 2 U 2 x+ x, into face : ρu 1 2 U 2, out of face + : ρu 1 2 U 2 +, into face : ρu 1 2 U 2, out of face + : ρu 1 2 U 2 +. With the expressions above, it is possible to rewrite the transport of the kinetic energ due to convection as: ρu x 12 U 2 x ρu x 12 U 2 x+ x + ρu 12 U 2 ρu 12 U 2 + x + ρu 12 U 2 ρu 12 U 2 + x. Defining the sum of additional source terms that act on the volume b the quantit e S and put the new evaluated terms into equation 2.55, we get: x t ρ1 2 U 2 = ρu x 12 U 2 x ρu x 12 U 2 x+ x + ρu 12 U 2 ρu 12 U 2 + x + ρu 12 U 2 ρu 12 U 2 + x +e S x. 2.58

42 2.4. THE CONSERVED MECHANICAL ENERGY EQUATION 35 To get to a partial differential equation, we will divide the whole equation b the volume of the element dv. Thus, we get: t ρ1 2 U 2 = ρu x 1 2 U 2 x ρu x 1 2 U 2 x+ x x + ρu 1 2 U 2 ρu 1 2 U ρu 1 2 U 2 ρu 1 2 U e S The next step is to use the assumption of equation 2.5 and 2.6. Therefore, we get the first part of the kinetic energ equation without an explicit mentioned source term: t ρ1 2 U 2 = x ρu 1 x 2 U 2 ρu 1 2 U 2 ρu 1 2 U 2 + e S The vector notation for this equation is: 1 t 2 U 2 = ρu 1 2 U 2 + e S Equation 2.61 can also be derived using the momentum equation and forming the scalar product of the local velocit and equation Source Terms of the Kinetic Energ The kinetic energ can be changed b several phenomena. These sources terms act on the volume element. To get the different source terms we will use the momentum equation The terms e S are simpl the source terms in the momentum equation 2.26 multiplied b the velocit. Recall: The source terms of the equation of motion are: Pressure surface, Shear-rate surface, Gravit volume.

43 36 Derivations of the Governing Equations Using this information, we can replace the quantit e S b: e S = τ U p U + ρg U work done b gravit Note: The multiplication with the velocit results in the inner product If we insert the source term e S into equation 2.61, we get the conserved mechanical kinetic energ equation with the source terms. 1 t 2 U 2 = ρu 1 2 U 2 τ U p U + ρg U Analing the new equation, we alread know the meaning of the term on the LHS and the the first and last term on the RHS. Thinking about the second and third term on the RHS is not so clear till now. Therefore, we will replace these terms b manipulating both b appling the product rule. The term that denotes the viscous force, can be rewritten as: [τ U] = τ : U + U τ gradient divergence double inner product inner product The pressure term can be manipulated like: Now we rearrange these equations: pu = U p +p }{{ U} gradient divergence inner product U τ = [τ U] τ : U, 2.66 U p = pu p U, 2.67 and insert both into equation The resulting equation gives us the possibilit to get a better phsical base for the meaning of the single terms. Hence, we get for

44 2.4. THE CONSERVED MECHANICAL ENERGY EQUATION 37 the source terms the following expression: e S = [τ U] work done b viscous force τ : U irreversibe conversion to internal energ shear heating p U + ρg U reversible conversion to internal energ work done b gravit pu work done b pressure of sourroundings Finall, we use the new evaluated sources e S to get a more understandable and readable kinetic energ equation: 1 t 2 ρ U 2 = ρu 1 2 U 2 [τ U] τ : U pu p U + ρg U The Meaning of Some Terms τ : U: as stated b Bird et al. [1960], this term is alwas positive for Newtonian fluids and describes that motion energ is irreversibl exchanged into thermal energ and therefore no real processes are reversible. This term will heat up the fluid internall. The heating due to this term will onl be measurable if the speed of the fluid is ver high large velocit gradients; e.g. high-speed flight or rapid extrusion. p U: this term will cool or heat the fluid internall due to sudden expansion or compression phenomena; e.g. turbines or shock-tubes Integral Form of the Conserved Mechanical Energ Equation The integral form of the kinetic energ equation 2.69 can be achieved b using the Gauss theorem 1.32 as before. Hence, we get: 1 t 2 ρ U 2 dv = ρu 1 2 U 2 nds [τ U] nds τ : UdV pu nds p UdV + ρg UdV. 2.70

45 38 Derivations of the Governing Equations Non-Conserved Mechanical Energ Equation As before, we are able to use the continuit equation, to change the conserved kinetic energ equation into the non-conservative form. Therefore, we have to split the time derivative and convection term using the product rule again. In addition we have to put the convective term to the LHS of equation Thus, the time derivative will change to: t ρ1 2 U 2 = ρ 1 t 2 U U 2 t ρ, 2.71 and the convection term to: ρu 12 U 2 = ρu 1 2 U U 2 ρu We end up with: ρ 1 t 2 U U 2 t ρ + ρu 1 2 U U 2 ρu = After we exclude ρ and 1 2 U 2 from the equations, we get: [ ρ t 1 2 U 2 + U 1 ] 2 U [ 2 U 2 ] t ρ + ρu = continuit As we can see, the second term on the LHS is ero due to continuit and therefore we get the non-conserved kinetic energ equation b using the definition of the total derivative 1.25: ρ D 1 2 U 2 Dt = [τ U] τ : U pu p U + ρg U Remark: It should be obvious that we can use equation 2.64 and 2.65 to change/eliminate some terms again.

46 2.5. THE CONSERVED THERMO ENERGY EQUATION The Conserved Thermo Energ Equation After we have the total energ and the kinetic energ equation, we can simpl get the equation for the thermo internal energ equation b subtracting equation 2.69 from To get the internal energ equation, we will split the time and convection term of equation 2.45 first. This is done to separate the single quantities. Hence, we get: ρe + 12 t ρ U 2 = ρue U t ρe + 1 t 2 ρ U 2 = ρue ρu 12 U The next step is to replace the underlined term b the conserved kinetic energ equation ρu t ρe 12 U 2 [τ U] τ : U pu p U + ρg U = ρue ρu 12 U 2 q + ρu g pu [τ U] + ρs We can see that the second, third, fifth and seventh term on the LHS cancel with the second, fourth, fifth and sixth term on the RHS. Note that ρu g = ρg U. Hence, we get the following equation for the kinetic internal energ equation: ρe τ : U p U = ρue q + ρs t After sorting the equation we get the final form: ρe = ρue t transport b convection τ : U irreversible energ b viscous dissipation shear heating p U reversible energ b compression q energ input b conduction +ρs heat source. 2.80

47 40 Derivations of the Governing Equations Integral Form of the Thermo Energ Equation The integral form of the internal energ equation 2.80 is observed using the Gauss theorem 1.32: t ρedv = ρue nds τ : UdV p UdV q nds + ρsdv Non-Conserved Thermo Internal Energ Equation As before, we can rewrite the conserved internal energ equation into a nonconserved form using the continuit equation. For that we split the time and convective terms again. As before, the terms look alwas similar. The time derivative will be manipulated to: and the convective term to: ρe = ρe t t + eρ t, 2.82 ρue = ρu e + e ρu After inserting the two terms into equation 2.80, we get: Now we extract ρ and e: ρ e t + eρ + ρu e + e ρu = t [ e ρ ] t + U e total derivative ρ +e [ ] t + ρu = continuit Due to the continuit equation, we can cancel out the second term and write the non-conserved internal energ equation as: ρ De Dt = τ : U p U q + ρs. 2.86

48 2.6. THE CONSERVED ENTHALPY EQUATION The Conserved Enthalp Equation The next equation that we are going to derive is the conserved enthalp equation. For that we will use the total energ equation 2.45 and the definition of the enthalp h, that is simpl the sum of the internal energ plus the kinematic pressure: h = e + p ρ If we replace e in equation 2.45 with the new expression, we get: { [ ρ h p ] + 12 } {[ t ρ ρ U 2 = ρu h p ] + 12 } ρ U 2 q + ρ U g pu [τ U] + ρs. For further simplification, we split the time and convective term: t ρh t ρp ρ + 1 t 2 ρ U 2 = ρuh ρu p ρu 12 ρ U 2 q + ρ U g pu [τ U] + ρs The terms that are underlined are equal and cancel out as well as the densit in the second term of the LHS. Thus, we get the conserved enthalp equation as: t ρh t p + 1 t 2 ρ U 2 = ρuh ρu 12 U 2 q +ρ U g [τ U] + ρs Integral Form of the Conserved Enthalp Equation The integral form of the conserved enthalp equation mechanical energ included is constructed b using the Gauss theorem. Hence, equation 2.90 can be rewritten like: t pdv + ρhdv t ρu 12 U 2 nds 1 t 2 ρ U 2 dv = ρuh nds + ρ U g dv q nds [τ U] nds + ρsdv. 2.91

49 42 Derivations of the Governing Equations Non-conserved Enthalp Equation As for each conserved equation, it is possible to change equation 2.90 to a nonconservative form b using the continuit equation. To manipulate the conserved equation, we first have to split the time and convection terms of the enthalp equation. Hence, the time derivation can be re-ordered as: The convection term will be manipulated to: t ρh = ρ t h + h t ρ ρuh = ρu h + h ρu Finall, we replace the split term in equation 2.90 and move the convective term to the LHS. The result is: ρ t h + h t ρ t p + 1 t 2 ρ U 2 + ρu h + h ρu = B taking out the densit ρ and enthalp h of the terms of interest: [ ] ρ t h + U h+ + h [ ] t ρ + ρu continuit + 1 t 2 ρ U 2 t p = and sort the equation, we get the non-conserved enthalp equation: ρ Dh Dt = 1 t 2 ρ U 2 + t p ρu 12 U 2 q + ρ U g [τ U] + ρs The Conserved Enthalp Equation onl Thermo In man literatures we find another enthalp equation. The difference is, that the mechanical energ is removed and we onl have the thermo energ included. Therefore, we need to subtract equation 2.90 with 2.69 and use 2.65: continues on next site { t ρh t p + } { 1 t 2 ρ U 2 t } 1 2 ρ U 2 =...

50 2.6. THE CONSERVED ENTHALPY EQUATION { ρuh ρu 12 } U 2 q + ρ U g [τ U] + ρs { ρu 1 2 U 2 [τ U] τ : U pu p U + ρg U} The outcome of the subtraction is, that a lot of terms can be canceled out. Furthermore, the 10th and 11th term on the RHS can be combined using the product rule. Thus, we are allowed to rewrite this term as U p. The equation we get is ver common and can be found in man literatures. t ρh p = ρuh q + ρs + τ : U + U p t Furthermore, it is possible to modif this equation b putting the second term of the LHS to the RHS: t ρh = ρuh q + ρs + τ : U + t p + U p Total derivative This lead to the the total derivative on the RHS for the pressure and we can appl the rule given b equation The modified equation is then given b: Dp ρh = ρuh q + ρs + τ : U + t Dt Note: The equation above can be found in the following literatures Bird et al. [1960], Feriger and Perić [2008], Schware [2013]. Keep in mind that we still did not introduce the definition of the shear-rate tensor τ. Therefore, we have a negative sign in front of τ.

51 44 Derivations of the Governing Equations

52 Chapter 3 The Governing Equations for Engineers Normall, it is sufficient enough for engineers to know the general conservation equation for an arbitrar quantit φ. Once the meaning of this equation is understood, we are able to probabl derive an kind of equation. The general governing conservation equation of an quantit φ is given b: t ρφ = ρuφ + D φ + S φ time accumulation convective transport diffusive transport source terms. 3.1 In the equation above, D stands for the diffusion coefficient, that can be a scalar or a vector and S φ stands for an kind of sources or sinks that influence the quantit φ. Now we are able to simpl derive the mass, momentum and other conservative equations out of this b replacing the quantit φ b the quantit of interest. 3.1 The Continuit Equation To derive the mass conservation equation, we have to replace φ b 1. Furthermore, we have to know that the mass is not transfered b diffusion and we suggest that the mass is not transfered into energ or vice verse; no source terms. Thus, we get the continuit equation 2.12: ρ = ρu. 3.2 t 45

53 46 The Governing Equations for Engineers Of course, if we have an incompressible fluid we get equation The Momentum Equation To get the momentum equation we replace φ b U. In addition, we need to know the diffusion term and all other source terms that influence the momentum in the volume element. The diffusion term determines the transport of momentum due to molecular effects τ. The source terms are: gravitational acceleration and the pressure force. Later on, we see a more general form of these equation. ρu = ρu U + τ p + ρg 3.3 t Note: As we see, the shear-rate tensor τ has a positive sign in this equation. In each equation before we had a negative sign. The sign change will be understood after we introduce the definition of the shear-rate components all components are negative. In the equation above we alread applied the definition of the shear-rate components and hence the sign has to change. The momentum equation shows, that it is possible to use the governing conservation equation to derive other more complex equations. Doing so, we alwas have to know which source terms are important and how the diffusion term looks like. If we get more familiar with the equations especiall with the stress-tensor and the source terms, it is ver eas to use this equation and derive the one that is needed. 3.3 The Enthalp Equation To derive the enthalp equation, we have to replace φ b h. This will lead to the internal energ equation. The diffusion term q can be expressed b the Fourier law q = λ T. In addition, the energ of a fluid can be changed b other sources like the pressure work, friction, and so on. These terms are given in the chapter before and are neglected now. t ρh = ρuh + λ T } +S {{ h } neglected. 3.4

54 3.3. THE ENTHALPY EQUATION 47 It should be mentioned that the enthalp equation has a special characteristic because is the necessit to know the temperature field T. The enthalp equation is of interest, if we are solving compressible fluids. That can be also analed from the OpenFOAM R toolbox. For incompressible fluids, no temperature or enthalp equation is used whereas for compressible fluid we solve the enthalp equation. However we set the temperature field, we recalculate the enthalp based on the temperature fluid and the fluid properties. Temperature equation The temperature equation can be derived using the thermodnamic relation: dh = c p dt. Assuming constant heat capacit and incompressibilit, we can manipulate the enthalp equation to get to the temperature equation: c p t ρt = c p ρut + λ T +S h neglected. 3.5 Depending on the field we are working on, we have to take care about the source terms. Example given: if friction, pressure work or the kinetic energ accumulation are reall influencing the enthalp equation, we have to take these phenomena into account. However, as alread mentioned above, the temperature equation was derived with the assumptions of incompressibilit and constant heat capacit. Therefore, we should be aware if the equation is valid in the case we are tring to solve. Solver crashes in OpenFOAM R are sometimes related to wrong coupled equations. One nice example would be, solving a fluid for incompressible fluids but using a temperature depended densit. It can be shown mathematicall that this case can cause troubles if the implementation is not done correct.

55 48 The Governing Equations for Engineers Common Source Terms Shear-heating viscous dissipation Shear-heating can be included to the enthalp equation. Therefore, we have to add the term that describes the shear-heating; compare equation 2.80: S sh = τ : U. 3.6 Pressure work Pressure can also increase the enthalp of a fluid during time. This can be expressed as; compare equation 2.90: S pw = p t. 3.7 Note: This term can be turned on and off in the enthalp equation b using the dpdt keword within the thermophsicalproperties file in OpenFOAM R. B default it is set to es. The term was introduced in OpenFOAM R version Additional pressure work There is also an additional pressure work done b the divergence of the pressure and velocit; compare 2.68: S apw = Up. 3.8 If we are dealing with incompressible fluids, we are allowed to sa that the additional pressure work is onl done b the gradient of the pressure p because we can split S apw using the product rule. If follows: S apw = Up = U p + p }{{ U} = U p. 3.9 continuit Other source terms can be found in chapter 2 or in the literature that was given at the beginning of this chapter.

56 Chapter 4 Summar of the Equations On the next site, all derived equations are given in a summar for a fast look-up. Depending on the problem we are focusing on, special terms can be neglected or has to be taken into account. Thus, we should be familiar with the toolbox which equation and which terms are solved and the equation respectivel. There are more equations that could be included here; for example the equation for solid mechanics stress calculation and/or magneto hdrodnamics Maxwellequations. Due to the fact that we did not discuss these special kind of equations, the are not presented here. 49

57 50 Summar of the Equations Table 4.1: Conserved equations for pure fluids Continuit ρ = ρu For incompressible fluids t we get U = 0 Momentum Forced Convection Free Convection t equation For τ = 0 we get the Euler ρu = ρu U τ ρβgt T0 Approximate; p = ρg t Bird et al. [1960] Total Energ ρe + 1 t 2 ρ U 2 = ρue U 2 q Sum of thermo and +ρ U g pu [τ U] + ρs mechanic energ Kinetic Energ t 1 2 ρ U 2 = ρu 1 2 U 2 [τ U] τ : U pu p U + ρg U mechanical energ Energ Internal Energ t ρe = ρue τ : U p U q + ρs thermo energ Enthalp ρh p = ρuh q + U p [τ U] + ρs h = e + p t t ρ Temperature ρcvt = ρucvt q τ : U T p t T ρ Dcp U + ρt Dt in terms of cvbird et al. [1960] Temperature ρcpt = ρucpt q τ : U + ln V t ln T ρ Dp Dt Dcp + ρt Dt in terms of cp Bird et al. [1960]

58 Chapter 5 The Shear-rate Tensor and the Navier-Stokes Equations The equations that we derived till now allow us to calculate the flow fields numericall. This enables the possibilit to get a better insight into the phsics and lead to a better understanding about the phenomena in the flow field which can be used to optimie designs or increase the efficienc of a special device. In addition, it is possible to extract quantities that are not measurable in realit just imagine a liquid metal and measuring the pressure or velocit in a simple wa. However, if we anale the equations we figure out that some quantities are not known; for example the shear-rate components τ ij. These unknown quantities have to be expressed b known one. The shear-rate tensor is expressed b different equations which depend on the behavior of the liquid. Here we distinguish between Newtonian and Non-Newtonian fluids. In this chapter we introduce the shear-rate tensor τ for Newtonian fluids. Further notes and information can be found in Feriger and Perić [2008], Bird et al. [1960], Dantig and Rappa [2009]. 51

59 52 The Shear-rate Tensor and the Navier-Stokes Equations 5.1 Newtonian Fluids If we investigating into Newtonian fluids, we use the Newtonian law for the shearrate viscous stress tensor τ. described as: It was shown that the nine components can be τ xx = 2µ u x x + 2 µ κ U, τ = 2µ u + 2 µ κ U, τ = 2µ u + 2 µ κ U, ux τ x = τ x = µ + u, 5.4 x u τ = τ = µ + u, 5.5 u τ x = τ x = µ x + u x. 5.6 The normal stress components have the quantit κ included. Bird et al. [1960] describes this variable as the bulk viscosit. However, in chapter 7, we see that this definition is not correct and we should describe the bulk viscosit in a different manner. Nevertheless, as Bird et al. [1960] mentioned, the quantit κ is not reall important for dense gases and liquids and can be neglected; again, later on we see wh. The other term in the normal stress components takes the value 2 3 µ, which man authors refers to the secondar viscosit, dilatation term or first Lamés coefficient that is again not correct. The main reason for that misleading definition is based on the fact that we can make the following correlation λ = 2 3 µ. Further information can be found in chapter 7 or directl in Gurtin et al. [2010]. However, we will keep κ for now in the equations. Remark In general it is sufficient to know how the shear-rate tensor is defined. However, if we are going to implement new models or establish new models, it is necessar to understand the different terms and their meaning. Therefore, it is a good choice to know wh and how we get to the shear rate tensor. In addition we should make clear statements about the quantities pressure and equilibrium pressure. Everthing about that can be found in Gurtin et al. [2010].

60 5.1. NEWTONIAN FLUIDS 53 Continued If we insert the above mentioned definitions of the shear-rate components into the momentum equations 2.23, 2.24 and 2.25, we will get the momentum equations for Newtonian fluids also known as Navier-Stokes equations. For the x-component of the Navier-Stokes equation we get: t ρu x = x ρu xu x + ρu u x + ρu u x { [ 2µ u ] x x x + 2 µ κ U [ u µ x + u ]} x p x + ρg x For the -component of the Navier-Stokes equation we get: [ ux µ + u ] x t ρu = x ρu xu + ρu u + ρu u { [ ux µ x + u ] + [ 2µ u ] x + 2 µ κ U 3 + [ u µ + u ]} p + ρg and finall the -component of the Navier-Stokes equation can be written as:. 5.7, 5.8 t ρu = x ρu xu + ρu u + ρu u { [ ux µ x + u ] + [ u µ x + u ] + [ 2µ u ]} + 2 µ κ U 3 p + ρg. 5.9 The three equations can be put together b using the Einsteins summation con-

61 54 The Shear-rate Tensor and the Navier-Stokes Equations vention: t ρu i = x j ρu j u i x i [ { 1 ui 2µ µ κ + u } j x j x i ui x i ] p x i + ρg i Introducing the deformation rate strain-rate tensor D, D = 1 2 [ U + U T ], 5.11 where T stands for the transpose operation, that means A ij A ji, we are able to write the vector form of this equation: [ ] 2 ρu = ρu U 2µD + t 3 µ κ UI p + ρg Finall, we take the negative sign into the brackets of the second term on the RHS to get the general Navier-Stokes equation in vector notation: t ρu + ρu U = 2µD + [ 23 ] µ + κ UI viscous stress tensor τ p + ρg The viscous stress tensor also named shear-rate or deformation rate tensor τ can be defined as: τ = 2µD + [ 23 ] µ + κ UI If we use κ = 0, which comes from the Stokes, we get the common Navier-Stokes equation as shown in almost all literatures: t ρu + ρu U = 2µD 23 µ UI p + ρg viscous stress tensor τ Another form of the momentum equation 5.13 can be achieved after pushing the

62 5.1. NEWTONIAN FLUIDS 55 pressure gradient into the viscous stress tensor: t ρu + ρu U = 2µD + {[ 23 ] } µ + κ U p I Cauch stress tensor σ +ρg The result of the bracket is called the Cauch stress tensor σ. A discussion about this quantit is given in chapter 6. Finall, it has to be mentioned that there are much more was to define this equation. Especiall if we distinguish between total pressure p and the equilibrium pressure p eq. See chapter 7. The momentum equation can also be defined for solids. Therefore we are using the Lamés coefficients. More information about wh we can use this equation for solids too, can be found in Gurtin et al. [2010]. constitutive relations are given in chapter 7. A short overview of some Implementation information and stabilit terms for solid stress simulations can be found in Jasak and Weller [1998] The Proof of the Transformation The following section discusses the transformation of the vector form into the Cartesian one. As before, we will investigate onl into the terms that are not discussed till now. The terms that we are going to investigate are the viscous stress tensor τ and the pressure gradient of equation µD + [ 23 ] µ + κ UI p. The first step is to split the terms: 2µD + [ 23 ] µ + κ UI p. It is eas to demonstrate that the pressure gradient is equal to the terms in equation 5.7, 5.8 and 5.9. Additionall it is eas to show that the expression of pi is equal to p. For that, we are using the mathematic operation 1.21: pi = p = x p p 0 =! 0 0 p p x p p = p. 5.17

63 56 The Shear-rate Tensor and the Navier-Stokes Equations As we can see, the terms are equal for the pressure. The investigation into the first term that includes the deformation rate tensor D, needs the mathematic operations 1.18 and 1.21: 2µD = 2µ 1 [ U + U T ] = µ = µ = µ x u x x u x u x u x x u x u x + ux x + u x + u x u x u u u x u u + u x u u x + u x + ux u + u u + u u x x u x u x u x u u u x u u u x + ux u + ux u + u T 5.19 u x u u = = x x x x 2µ u x x µ[ ux µ[ u x µ[ ux x + ux x ] µ[ ux + u x ] µ[ ux + u x ] + + u x ] + + u x ] + u µ[ x + ux ] u µ[ + u ] u µ[ + u ] µ[ u x + ux 2µ u u µ[ x + ux ] u µ[ + ux ] u µ[ + u ] ] + µ[ u + µ[ u µ[ u + u ] + ] x + ux ] + ux 2µ u 5.22! = of x mom. of mom. of mom Now we need to check if the terms are correct; of course equation 5.23 alread shows that the terms have to be similar but we will demonstrate wh. For that, we use the term within the brackets {...} of equation 5.7, { [ 2µ u x x x ] µ κ U [ µ ux + u x ] + [ u µ x + u ]} x, 5.24.

64 5.1. NEWTONIAN FLUIDS 57 taking the minus sign into the brackets, split the x-derivative, we end up with: 2µ u x 23 x x x µ κ U + [ ux µ + u ] + [ u µ x x + u ] x Appl the same procedure on the terms of equation 5.8 and 5.9, we get the analed term of the -momentum to: [ ux µ x + u ] + 2µ u x 23 µ κ U + [ u µ + u ], 5.26 and the term of the -momentum is equal to: [ ux µ x + u ] + [ u µ x + u ] + 2µ u 23 µ κ U The underlined terms occur in equations That means, that the vector form and Cartesian one are similar for these term but there is one term missing in each derivative. This term comes from the last term that we neglected till now. The last term can be manipulated to: [ ] 2 3 µ κ UI [ ] = 3 µ κ U [ 2 x 3 = µ κ] U 0 0 [ µ κ] U [ µ κ] U x [ 2 3 µ κ] U of x mom = [ 2 3 µ κ] U =! of mom [ 2 3 µ κ] U of mom

65 58 The Shear-rate Tensor and the Navier-Stokes Equations As demonstrated, all terms of the shear-rate tensor are similar and therefore the vector form is identical to the three single Cartesian ones The Term 2 µ U 3 As mentioned at the beginning of this chapter, the second term 2 3 µ is referred in man literatures to be the dilatation viscosit. As alread said, the choice of this nomenclature is a bad one. However, this term represents expansion and compression phenomena. To demonstrate the meaning of the term and the correlation to expansion and compression phenomena, we will use the continuit equation to modif this equation. The mass conservation equation is given b: After appling the product law 1.22, we get: ρ + ρu = t ρ + ρ U + U ρ = 0, t 5.32 U = 1 [ ] ρ ρ t + U ρ This expression is now insert into the shear-rate tensor τ of equation The outcome is the following: We clearl see that the second term on the RHS is onl related to the densit change and thus, it is related to expansion and compression phenomena: τ = 2µD 2 { 3 µ 1 [ ]} ρ ρ t + U ρ I expansion and compression Further Simplifications If we assume incompressibilit of the fluid, ρ = constant, we can use the continuit equation for a simplifing the shear-rate tensor. Hence, the second term in equation 5.34, the dilatation term, depends onl on the densit, this term will vanish based on the fact that the densit will not change during time and the gradient of a constant number is ero. In addition, we are allowed to take out the densit of all remaining derivatives. Thus, we can divide b this quantit to get rid of the

66 5.1. NEWTONIAN FLUIDS 59 densit. The result of the shear-rate tensor is as follows ν = µ ρ : τ = 2νD If the dnamic viscosit ν can be assumed as constant, we further can simplif the equation b taking out the viscosit of the divergence operator: τ = ν U + U T The underlined term results in a tensor that can be simplified b the continuit equation. Thus, we get the famous Laplace equation: τ = ν 2 U = ν U. 5.37

67 60 The Shear-rate Tensor and the Navier-Stokes Equations

68 Chapter 6 Relation between the Cauch Stress Tensor, Shear-Rate Tensor and Pressure In equation 5.16 we introduced the Cauch stress tensor σ. This stress tensor includes all stresses that act on the volume element dv. That means, shear and pressure forces because both can related to stresses. How the total stress, shear-rate stress and pressure are related is briefl discussed in this chapter. First we start with the introducing of the Cauch stress tensor σ: σ xx σ x σ x σ = σ x σ σ. 6.1 σ σ σ As we know from chapter 1, we are able to split each matrix into a deviatoric and hdrostatic part. The definition is given in equation 1.28 and can be applied to the Cauch stress tensor: σ = σ hd + σ dev. 6.2 The hdrostatic part of an arbitrar matrix A has the special meaning of the negative pressure p. Hence, equation 1.29 or 1.30 can be related to the pressure as: p = A hd = 1 tra,

69 62 Relation between the Cauch Stress Tensor, Shear-Rate Tensor and Pressure pi = A hd I = 1 trai Using the definition of the deviatoric part 1.31 and the above expression, we can rewrite equation 6.2: σ = pi + [σ 13 ] trσi. 6.5 shear rate tensor τ The deviatoric part is defined as the shear-rate stress and can be expressed b the shear-rate tensor τ : σ dev = τ, 6.6 σ = pi + τ. 6.7 Depending on the fluid we are using, τ has to be replaced with the correct expressions. Note: In fluid dnamics it is common to use the split Cauch tensor to get the shear-rate tensor and pressure. In solid mechanics it is common to work with the full stress tensor σ, compare Jasak and Weller [1998].

70 Chapter 7 The bulk viscosit As we alread pointed out in chapter 5, Bird et al. [1960] referred the quantit κ which is included in the normal components of the shear-rate tensor as bulk viscosit. The bulk viscosit is also named, dilatation viscosit, volumetric viscosit or volume viscosit. All of these names represent the same. In order to make a clear statement about the bulk viscosit, we have to go deeper into the basics of the mechanics and thermodnamics. All necessar informations can be found in Gurtin et al. [2010] which are summaried and compressed in the following chapter for compressible fluids that show a linear function in the viscosit for the shearing. First we have to know that an isotropic linear tensor function TA is isotropic, if and onl if there are two quantities scalars µ and λ do not mix with first and second Lamé coefficient or molecular viscosit here such that we can define the tensor function as: TA = 2µA + λ trai. 7.1 Gurtin et al. [2010] proofed that the shear-rate tensor also named stretch tensor in their book is an isotropic tensor which can be shown using frame-indifferences and thus τ has to follow the equation above. To demonstrate which quantit defines the bulk viscosit, we have to start with the Cauch stress tensor σ which is depended on the densit and the strain-rate tensor D. Wh we have the densit included can be seen in equation 7.9. It follows: σ = σρ, D. 7.2 Defining a new quantit σρ, 0 that represents the stress in the fluid in the absence of flow no velocit gradients and therefore no shearing, we are able to write the 63

71 64 The bulk viscosit viscous deviatoric part of the Cauch stress tensor as: σ vis ρ, D = σρ, D σρ, Based on the fact that σρ, 0 has to be an isotropic tensor, it has the specific form to be equal to the negative equilibrium pressure p eq : σρ, 0 = p eq ρi. 7.4 Finall, after we sum up the latest results, we end up with: σρ, D = σ vis ρ, D p eq ρi. 7.5 Equation 7.5 is similar but not equal to 6.7 based on the the equilibrium pressure and the viscous term. Later we see wh both equations are equal for some particular cases in which we are working commonl. Using the alread known definition of the total pressure p, which is given b: pρ = 1 trσ, we are able to express the total pressure b the equilibrium pressure and the viscous stress tensor: pρ = 1 3 tr [ p eq ρi + σ vis ρ, D ]. 7.7 It is trivial that this lead to: pρ = p eq ρ 1 3 tr [σ visρ, D]. 7.8 Now it is obvious that the total pressure for compressible fluids is based on two different contributions, the equilibrium pressure and a part, namel the isotropic one, of the viscous part which is based on internal friction. The question that ma arise is: Where is the equilibrium pressure in all equations that we had before. To answer this, we have to use the relation 7.1 for the viscous part, σ vis ρ, D. Thus, we get: σ vis ρ, D = 2µρD + λρ trdi. 7.9 The next step is to split the deformation strain-rate tensor in its deviatoric and

72 65 hdrostatic spherical part: D = D trdi, and insert this expression into the first term on the RHS of equation 7.9. The quantit D 0 represents the deviatoric part of the strain-rate tensor which include onl friction which acts angular. In other words, these components tr to deform the volume element while keeping the volume of the element constant no compression or expansion. Thus, it follows: σ vis ρ, D = 2µρ [D ] trdi + λρ trdi, 7.11 = 2µρD 0 + 2µρ 1 trdi + λρ trdi, 7.12 [ 3 ] 2 = 2µρD 0 + µρ + λρ trdi, =κρ = 2µρD 0 + κρ trdi κ represents the bulk viscosit, volumetric viscosit, volume viscosit or dilatation viscosit and is equal to: κρ = 2 µρ + λρ Now we stated the correct bulk viscosit definition. Comparing this with Bird et al. [1960], we see that there is a difference in the nomenclature. Now, if we take the trace of equation 7.14: and put this into equation 7.8, we get: and thus the Cauch stress tensor can be written as: tr [σ vis ρ, D] = 3κρ trd, 7.16 pρ = p eq ρ κρ trd, 7.17 σρ, D = 2µρD 0 pρi Again, this equation looks similar to equation 6.7 but is in fact not equivalent.

73 66 The bulk viscosit In addition the above constitutive equation for the stress tensor σ is onl valid for compressible fluids which have a linear viscosit behavior. To get the equivalence of both equations we have to go further. As Bird et al. [1960] alread mentioned, the quantit κ is not too important and can be neglected which is also stated b Gurtin et al. [2010]. Based on that, it is common to use the Stokes relation given b the definition κ = 0. Doing that, we directl see that λ has to be equal to 2 3 µ, cf If we insert the outcome related to the Stokes hpothesis into equation 7.9, and express the quantit µ with the molecular viscosit of the fluid, we get the expression for the shear-rate tensor τ ; cf. equation In addition we see, considering equation 7.17, that the total pressure p is equal to the equilibrium pressure p eq in the case when we assume the Stokes relation. Thus, both equations are equivalent. Remark In this chapter we used a form of the equations which included the functionalit dependenc of the quantities. This was done just in order to show the dependenc which is not reall necessar. In addition, this was just a brief summar for compressible linearl viscous fluids. Furthermore, keep in mind that the variables λρ and µρ in this section are not related to the molecular viscosit or the Lamés coefficients and represent two arbitrar scalar functions which depend on the densit. Further information can be found in Gurtin et al. [2010].

74 Chapter 8 Collection of Different Notations of the Momentum Equations In literatures we find a lot of different notations for the momentum equation for Newtonian fluids. It is obvious that we can change the stress tensors as we want and pla around with the mathematic laws to manipulate the equation as we like to have it. Common conserved momentum equation 2.26: ρu = ρu U τ p + ρg. 8.1 t Conserved momentum equation with the shear-rate tensor: ρu = ρu U + t Conserved momentum equation as used in chapter 7: ρu = ρu U + t Conserved momentum equation with bulk viscosit: 2µD 23 µ UI p + ρg µD 0 + λ + 23 µ UI p + ρg. 8.3 t ρu = ρu U + 2µD 0 + κ UI p + ρg

75 68 Collection of Different Notations of the Momentum Equations Conserved momentum equation with trace operator: ρu = ρu U + 2µD 23 t µ trdi p + ρg. 8.5 General conserved momentum equation with Cauch stress tensor: ρu = ρu U + σ + ρg. 8.6 t Non-conserved momentum equation with Cauch stress tensor: ρ DU Dt Integral form of momentum equation: ρudv = ρu U nds τ nds t = σ + ρg. 8.7 pi nds + ρgdv. 8.8 It should be obvious that we can transform all equations above into the nonconservative or integral form. The Proof that the Trace Operator replaces the Divergence Operator In one equation above we replaced the divergence operator b the trace operator. That the operation trd results in U is shown now: The demonstration is ver simple: U = trd [ trd = tr U + U T ] 2 = 1 [ 2 u ] x 2 x + 2u + 2u = U The divergence operator is evaluated b equation 1.20.

76 Chapter 9 Turbulence Modeling In this chapter we focus on turbulent flow fields and the Renolds-Averaging approach which was introduced b Osborne Renolds. First we will investigate into different averaging approaches. Then we are going to derive the incompressible mass and momentum equation to show the closure problem. After that, we discuss some hpothesis that are used to get rid of the closure problem. That lead to the derivative of the Renolds stress equation. The outcome of this equation is the anale of the analogies to the Cauch stress tensor and the derivation of the turbulent kinetic energ. Finall, we discuss the main problem if we want to average the compressible mass, momentum and energ equation and introduce the Favre averaging concept. The main literature that is used in this chapter is Feriger and Perić [2008], Bird et al. [1960], Wilcox [1994]. 9.1 Renolds-Averaging The investigation into flow fields are generall turbulent and hence, it is a challenging task to resolve the flow with all details in other words, with all phsics. Observing an arbitrar flow field, figure 9.1, we can anale that the flow has a deterministic character. That means, the flow is chaotic and can be prescribed using a time independent mean value φ and its fluctuation φ that is oscillating around the mean value. This behavior is valid for each quantit we focus on like u x, u, u, T, h, c and so on; for figure 9.1, φ would be the velocit u in one direction and could be expressed as: φt, x = φx + φ t, x

77 70 Turbulence Modeling now: Osborne Renolds introduced several averaging concepts that are presented Time averaging, Spacial averaging, Ensemble averaging. The time averaging method can be used for a statistic stationar turbulent flow left figure of 9.1. Defining the instantaneous flow variable b φt, x and the time averaged one b φx T, the concept is defined b: 1 t+t φx T = lim φt, xdt. 9.2 T T t In order to get good results, T should be chosen large enough compared to the time scale of the fluctuation φ. That is the reason wh we are interested in the case when T goes to. As we observe in the figure and in the equation, the averaged value is no longer time depended. The spacial averaging method is appropriate for homogeneous turbulent flows. This ields to a uniform turbulence in all space directions. Here we average over the volume. Renaming the averaged quantit to φx V, we can write: 1 φx V = lim V V φt, xdv. 9.3 The ensemble averaging method is the most general method. Think about a series of measurement with the number of N identical experiments where φ n t, x = φt, x at the n th series. The concept can be defined as follows; the averaged value is denoted b φt, x E : 1 φt, x E = lim N N N φ n t, x. 9.4 For turbulent flow fields that are stationar and homogeneous, all three concepts are similar and lead to the same result. This is also known as the ergodic hpothesis. The averaging method that we choose for the further investigation is the time averaging method. The reason for that is based on the fact that things can be described eas and clear. Let us focus on the left part of figure 9.1 first. B n=1

78 9.1. REYNOLDS-AVERAGING 71 Mean value Real value Mean value Real value Velocit Velocit T1 T T2 N N Time Time Figure 9.1: Averaging of a statistic stationar left and statistic non-stationar flow right. replacing the instantaneous variable in equation 9.2 b the definition 9.1, we get: 1 t+t [ φx = lim φx + φ t, x ] dt. 9.5 T T t We observe that the time averaging of the mean and fluctuation quantit leads to the mean quantit again. Thus we can show, that the time average of an alread averaged mean quantit is the mean quantit again: 1 t+t φx = lim T T t φxdt. 9.6 In addition we can see that the time averaging of the fluctuation is ero. demonstrate that, we use equation 9.1 and replace the mean quantit on the RHS of equation b 9.6: 1 t+t lim φ 1 t+t [ ] t, xdt = lim φt, x φx dt T T t T T t 1 t+t 1 = lim φt, xdt lim T T T T 1 = lim T T t t+t t t+t [ φx + φ t, x ] dt lim T t φxdt 1 T t+t t To φxdt = φx φx = Some remarks of the validit of this method can be found in Wilcox [1994]. If we think about flows where the mean value of the instantaneous quantit is

79 72 Turbulence Modeling changing during time non-stationar flows, figure 9.1; right, we have to modif equation 9.1 and 9.2 to: φt, x = φt, x + φ t, x, t+t φt, x = lim φt, xdt. 9.9 T T t Some remarks and limits to equation 9.9 are given in Wilcox [1994]. Correlation for the Renolds-Averaging For the derivations of the Renolds-Averaged conservation equations, we need some mathematic rules. For now we will use an overline to indicate that we use the Renolds time-averaging concept instead of writing the integrals. Linear Terms Appling the average method 9.9 to linear terms lead to linear averaged terms. To demonstrate this, we will use two linear functions ft, x and gt, x: ft, x = 1 T t+t t ft, xdt = 1 T t+t t [ ft, x + f t, x ] dt = ft, x + f t, x = ft, x ft,x =0 gt, x = 1 T t+t t gt, xdt = 1 T t+t t [ḡt, x + g t, x ] dt = ḡt, x + g t, x = ḡt, x ḡt,x =0 Note: The result can be time dependent or not. This behavior is related to the problems we are looking at. For stationar problems we will end up with fx, ḡx, whereas for non-stationar problems we get the terms we derived above.

80 9.1. REYNOLDS-AVERAGING 73 If we have the sum of two linear terms ft, x + gt, x, we get: ft, x + gt, x = 1 T = 1 T = 1 T t+t t t+t t t+t t [ft, x + gt, x] dt ft, xdt + 1 T = ft, x + f t, x t+t t gt, xdt [ ft, x + f t, x ] dt + 1 T t + ḡt, x + g t, x t+t [ḡt, x + g t, x ] dt = ft, x + ḡt, x Averaging linear terms end up with the same linear terms but now with the mean quantities. The fluctuation terms will vanish based on the proof we did above. Non-Linear Terms If we focus on non-linear terms like ft, xgt, x, we produce new additional terms during the averaging procedure. To demonstrate this, we will use the above term and average it. What we get is: ft, xgt, x = 1 T = 1 T = 1 T t+t t t+t t t+t t [ft, xgt, x] dt [{ ft, x + f t, x } { ḡt, x + g t, x }] dt [ ft, xḡt, x + ft, xg t, x +f t, xḡt, x + f t, xg t, x ] dt = ft, xḡt, x + ft, xg t, x + f t, xḡt, x + f t, xg t, x Rewriting the whole equation, we get: ft, xgt, x = ft, xḡt, x + f t, xg t, x additional terms 9.12 The reason wh the second and third term cancels out is due to the fact that the fluctuation is linear in this terms and hence equation 9.7 is valid. For the last term, there is no reason for the product of the fluctuations to vanish.

81 74 Turbulence Modeling Constants Finall, the Renolds time-averaging concept does not affect constant quantities. Defining an arbitrar constant a, we get: aft, x = 1 T = a 1 T t+t t t+t t aft, xdt = a ft, x + a f t, x [ ft, x + f t, x ] dt = a ft, x Renolds Time-Averaged Equations The Navier-Stokes equations give us the possibilit to resolve each vortex and hence all flow phenomena. Appling the equations to turbulent flow fields is a hard and challenging topic that requires extreme fine meshes and time steps and lead to high computational costs. Furthermore, engineers are commonl onl interested in some averaged values and on some special phsics. Hence, there is no need to resolve all details. Thus, we use the Renolds-Averaging concept to simplif the flow equations; in other words, the whole turbulence behavior is approximated with models. Considering incompressibilit, the Renolds time-averaged equations can be derived relativel easil compared to compressible flow fields. This will be discussed using the mass conservation equation now Incompressible Mass Conservation Equation The start point for the derivation is the compressible mass conservation equation 2.12 in the form of Cartesian coordinates: ρ t + ρu x + ρu + ρu = x Assuming incompressibilit of the fluid, we are allowed to put the densit out of the derivatives; it is obvious that the time derivative will vanish and we end up with: ρ u x x + ρu + ρu =

82 9.2. REYNOLDS TIME-AVERAGED EQUATIONS 75 Replacing the values u i b the assumption 9.1, ρ ū x + u x x + ρ ū + u + ρ ū + u = 0, 9.16 and appl the Renolds time-average concept, ρ ū x + u x x + ρ ū + u + ρ ū + u = 0, 9.17 we get the time-averaged incompressible mass conservation equation: ρ ū x x + ρū + ρū = ρ Ū = Of course we are allowed to divide the whole equation b the densit Compressible Mass Conservation Equation Doing the same average procedure with the compressible mass conservation equation lead to a more complex form because we also have to consider the densit as a varing quantit. If follows: ρ t + ρu x + ρu + ρu = 0, 9.19 x ρ + ρ t + [ ρ + ρ ū x + u x] x + [ ρ + ρ ū + u ] + [ ρ + ρ ū + u ] = 0, 9.20 ρ + ρ t + [ ρ + ρ ū x + u x] x + [ ρ + ρ ū + u ] + [ ρ + ρ ū + u ] = 0, 9.21

83 76 Turbulence Modeling ρ t + ρū x + ρu x + ρ ū x + ρ u x x + ρū + ρu + ρ ū + ρ u ρu + + ρū + ρ ū + ρ u = 0, 9.22 ρ t + ρū x + ρ u x + ρū + ρ u + ρū + ρ u = 0, 9.23 x ρ t + ρū i + ρ x u i = i Introducing a vector that contains the fluctuations of the velocities u x, u, u, U = u x u u, 9.25 we can rewrite the Renolds time-averaged compressible mass conservation equation in vector notation: ρ t + ρū + ρ U = Due to the fact that we have two quantities that have to be averaged, we get non-linear terms, that lead to new unknown ρ u i. Due to this behavior, we will investigate into the incompressible flows fields first. Averaging the compressible equations will be discussed in section Incompressible Momentum Equation The derivation of the time-averaged x-component of the momentum equation will be discussed now. For the derivation, we will use equation 5.7 and assume incompressibilit of the fluid. For the and components, the equations 5.8 and 5.9 have to be used. Due to the fact that the derivations are identical, we onl give the final equation for and without all steps. The x component is now analed and averaged in detail.

84 9.2. REYNOLDS TIME-AVERAGED EQUATIONS 77 x-component of Momentum The incompressible momentum equation for the x-component is given b: ρ t u x = ρ x u xu x + ρ u u x + ρ u u x { [ 2µ u ] x + [ µ x x + [ u µ x + u x ux + u x ]} p x + ρg x. ] 9.27 This is the start point for the procedure. The first step is to replace the velocit quantities b equation 9.1 and appl the Renolds time-averaging concept. For clearance, we will examine each term separatel. Starting with the term on the LHS, we get: The first term on the RHS ends up as: ρ t ū x + u x = ρ tūx ρ x ū x + u xū x + u x + ρ ū + u ū x + u x + ρ ū t + u t ū x + u x To simplif the terms, we focus on each term separatel. Therefore we get: ρ x ū x + u xū x + u x = ρ [ūx ū x + x ū x u x + u xū ] x + u xu x = ρ x ū xū x + ρ x u xu x, ρ ū + u ū x + u x = ρ [ ū ū x + ū u x + ] u ū x + u u x = ρ ρū ū x + ρ u u x, 9.30 ρ ū + u ū x + u x = ρ ] [ū ū x + ū u x + u ū x + u u x = ρ ρū ū x + ρ u u x. 9.31

85 78 Turbulence Modeling Finall, the first term on the RHS can be written as: ρ x ū xū x + ρ x u xu x + ρ ū ū x +ρ u u x + ρ ū ū x + ρ u u x. After sorting the terms, we end up with: ρ x ū xū x + ρ ū ū x + ρ ū ū x identical convective terms ρ x u xu x + ρ u u x + ρ u u x additonal terms; Renolds Stress. The second term on the RHS of equation 9.27 will be discussed now. The term is given b: { [ 2µ u ] x + [ ux µ x x + u ] + [ u µ x x + u ] } x Term 1 Term 2 Term 3 B analing term one to term three step b step, we get for the first one the following expression: for the second one this expression: [ 2µ ū x + u ] x = [ 2µ ū ] x x x x x [ ūx + u µ x + ū + u ] = [ ūx µ x + ū ] x and for the third term this one: [ ū + u µ + ū x + u ] x = [ ū µ x x + ū ] x, 9.32, Finall, after combining all three parts, we get the following expression for the

86 9.2. REYNOLDS TIME-AVERAGED EQUATIONS 79 second term on the RHS of equation 9.27: { [ 2µ ū ] x + [ ūx µ x x + ū ] + [ ū µ x x + ū ]} x After we analed the first two terms, we will investigate into the last two terms on the RHS of equation It follows: p x + ρg x = p + p + ρg x = p x x + ρg x Now we can rewrite the x-component of the momentum equation 9.27 as Renolds time-averaged x-momentum equation: ρ = ρ tūx xūxū x + ρ ūū x + ρ ūū x ρ x u xu x + ρ u u x + ρ u u x { [ 2µ ū ] x + [ ūx µ x x + ū ] x + [ ū µ x + ū ]} x p x + ρg x If we appl the same procedure to the and component of the momentum equation, we get the Renolds time-averaged momentum equation for the and components respectivel. These three Renolds time-averaged equations are then called Renolds-Averaged-Navier-Stokes equations RANS.. -Component of Momentum ρ = ρ tū xūxū + ρ ūū + ρ ūū ρ x u xu + ρ u u + ρ u u { [ 2µ ū ] + [ ūx µ x + ū ] x + [ ū µ + ū ]} p + ρg. 9.37

87 80 Turbulence Modeling -Component of Momentum ρ = ρ tū xūxū + ρ ūū + ρ ūū ρ x u xu + ρ u u + ρ u u { [ 2µ ū ] + [ ūx µ x + ū ] x + [ ū µ + ū ]} p + ρg 9.38 If we are using the vector of fluctuations U 9.25, the definition of the deformation strain rate tensor D 5.11 and taking the convective terms to the LHS, we can rewrite the averaged momentum equation in vector form as: ρ tū + ρ Ū Ū = 2µ D p + ρg } {{ τ } Same as equation 5.13 with ρ=const. ρ U U Renolds Stresses σ t D defines the Renolds-Averaged mean deformation rate tensor, τ the mean shear-rate tensor and the last term the Renolds-Stresses, denoted as Renolds- Stress tensor σ t ; in man literatures we will find the greek smbol τ t to express the Renolds-Stress tensor this is omitted here because otherwise we are not able to show the analogies between the real stress tensor σ Cauch stress tensor and the Renolds-Stress tensor σ t clearl. The Renolds-Stress tensor σ t is defined as: ρu xu x ρu u x ρu u x σ t = ρu i u j = ρu xu ρu u ρu u = σ txx σ tx σ tx σ tx σ t σ t. ρu xu ρu u ρu u σ tx σ t σ t 9.40 After we introduced the Renolds-Stress tensor, we can rewrite the momentum equation in a more general form: ρ tū + ρ Ū Ū = τ p + ρg + σ t Finall, we will use the relation between the Cauch stress tensor, the shear-rate

88 9.2. REYNOLDS TIME-AVERAGED EQUATIONS 81 tensor and the pressure 6.7. Hence, we end up with the following equation: ρ tū + ρ Ū Ū = σ + ρg + σ t In section 2.2 we alread showed and discussed that the vector form results in the Cartesian one. In the above equation there is onl one term left that we should transformed to demonstrate that each term of the vector form represents the corresponding term in the Cartesian equation. Hence, we will onl investigate into that one. The Renolds-Stress term can be rewritten as: ρ U U = ρ = [ [ ρ [ ρ x u x u x + ρ x ρ x u x u u u x u u 9.43 x u xu x u xu u xu = ρ u u x u u u u 9.44 u u x u u u u u x u + ρ u x u ] of x mom. u u x + ρ x u u + ρ x u u ] u u x + ρ x u u + ρ u u ] =! of mom of mom. As we can see and alread knew, the terms are equal. In most literatures we will find the Renolds time-averaged momentum equations in Cartesian form using the Einsteins summation convention. This lead to the following equation: ρ + ρ ū i ū j = τ ij p + ρg i ρ u tūi i x j x j x i x u j j Furthermore, sometimes the Renolds-Stress term is put into the convective term on the LHS. Hence, we get: ρ + ρ ū i ū j + u tūi i x u j = τ ij p + ρg i j x j x i The derivation of the Renolds-Averaged momentum equations are done. boxed equations above are known as Renolds-Averaged-Navier-Stokes equations RANS. The

89 82 Turbulence Modeling Note: It should be obvious that we can put the densit in or out of the derivatives it is a constant if we use the assumption of incompressibilit. Hence, this formulation is also valid: t ρū i + ρū i ū j + ρu i x u j = τ ij p + ρg i j x j x i In addition it is clear, that we are allowed to divide the equations b the densit ρ. For that, we just have to be sure to have the right quantities for the pressure and the dnamic viscosit µ. The dnamic viscosit will become the kinematic viscosit ν and the pressure is divided b the densit. Furthermore we can think about the gravitational acceleration term ρg i. If the densit is constant, this term gets constant and can be neglected because it will not change the momentum in an case. If we still want to have a buoanc term within the incompressible equations, we need to use some models like the Boussinesq approximation The Incompressible General Conservation Equation After we derived the RANS equations, the derivation of all other conserved equations like the enthalp, temperature or species equation can be done with the same procedure but is not demonstrated now. As we alread know, we could use a general conservation equation to derive other equations. Therefore, we will derive the Renolds time-averaged governing conserved equation 3.1 for incompressible fluids without an source terms. Hence, the starting point is: ρ t φ = ρ Uφ + D φ time accumulation convective transport diffusive transport To show the transformation to the Renolds time-averaged equation, we will switch this equation into the Cartesian form first b using the mathematics 1.20 and assume that the diffusion coefficient D represents a vector; like different thermal diffusivit coefficients in the three space directions otherwise the derivation get

90 9.2. REYNOLDS TIME-AVERAGED EQUATIONS 83 simplified and is not worth do show: ρ t φ = ρ x u xφ + ρ u φ + ρ u φ + x D x φ x + D φ + φ D The next step is to use the expression from equation 9.1 and appl it to φ, u x, u and u keep in mind that D are constant values and are not influenced b the averaging procedure. It follows: ρ t φ + φ = ρ x [ ū + u φ + φ ] + + ρ + [ ūx + u x φ + φ ] + ρ D φ + φ + [ ū + u φ + φ ] D x x D φ + φ x φ + φ To discuss the Renolds-Averaging procedure, we will anale each term separatel of equation Therefore, the time term results in: The first term on the RHS, ρ t φ + φ = ρ t φ ρ [ ūx + u x x φ + φ ] + ρ [ ū + u φ + φ ] +ρ [ ū + u φ + φ ], will be split to enable analing term b term. It follows: ρ [ ūx + u x x φ + φ ] = ρ [ ū x φ + ūx φ x + u φ ] x + u xφ = ρ x ū x φ + x ρu xφ, 9.53

91 84 Turbulence Modeling ρ [ ū + u φ + φ ] = ρ [ ū φ + ū φ + u φ ] + u φ = ρ ū φ + ρ u φ, 9.54 ρ [ ū + u φ + φ ] = ρ [ ū φ + ū φ + u φ ] + u φ Hence, the first term on the RHS after sorting is: = ρ ū φ + ρ u φ ρ x ū x φ + ρ ū φ + ρ ū φ + ρ x u xφ + ρ u φ + ρ u φ The second, third and fourth term on the RHS will end up as: D x x x φ + φ = D φ x x x D φ + φ = D φ + φ = D φ D φ, 9.56, To sum up, the general Renolds-Averaged conservation equation can be written as: ρ t φ = ρ and is given in vector form b: x ρū x φ + ρ ū φ + ρ ū φ + D φ x + D φ + D φ x x ρ x u xφ + ρ u φ + ρ u φ turbulent scalar flux ρ t φ = ρ Ū φ + D φ ρ U φ same as before turbulent scalar flux,

92 9.3. THE CLOSURE PROBLEM 85 Equation 9.60 allows us to derive each Renolds-Averaged conservation equation b replacing φ with the quantit of interest analog to chapter 3. As we alread realied, after deriving the Renolds-Averaged momentum equations, we get the same equations but with additional terms. These additional terms are called Renolds-Stresses for the momentum equations and are named additional turbulent scalar flux for all other quantities. Finall and if we want, we can put the densit inside the derivations and we end up with: t ρ φ = ρū φ + D φ ρu φ same as before turbulent scalar flux Manipulating the equations should be familiar now. Thus, we can put the convective and the turbulent scalar flux terms together. In addition we add the arbitrar source term of φ. The resulting general Renolds-Averaged conservation equation is then written as: t ρ φ + ρū φ + ρu φ = D φ + S φ The Closure Problem The Renolds-Averaged procedure lead to the problem, that we create additional unknown quantities and no further equations. In other words, the terms ρu i u j and ρu i φ are not known and cannot be calculated. Hence, the set of equations are not enough to close our problem and we cannot solve our sstem. This is known as closure problem. Therefore, we need approximations that correlate the unknown with known quantities. Till toda this problem is still not solved and we do not have a set of equations to get rid of the closure problem and therefore, we are forced to use approximations, if we use the Renolds time-averaging procedure. The equations that are introduced b authors to get rid of the closure problem are known as turbulence models. Within this assumptions, we tr to correlate the unknown quantities with known one. For the Renolds-Stresses and turbulent scalar fluxes we can use several theories that tr approximate the unknown terms. The most popular methods are the Boussinesq s edd viscosit, the Prandtl s mixing length or the Von-Kármán s

93 86 Turbulence Modeling similarit hpothesis. Further information about these theories concept of higher viscosit can be found in Feriger and Perić [2008], Bird et al. [1960], Wilcox [1994]. Kewords: energ cascade, higher viscosit concept, edd viscosit, dissipation and turbulent viscosit. 9.4 Boussinesq Edd Viscosit The most used hpothesis is the theor postulated b Joseph Boussinesq that simpl relates the turbulence of a flow to a higher fluid viscosit. The thought behind is as follows: If we have a higher turbulence flow, the flow gets more chaotic and we get a lot of vortexes that can transport for example heat in addition to the alread existing transport phenomena. Therefore, it is clear and of humans nature to sa, that we could achieve that, if we increase the diffusion coefficient the viscosit in the momentum equation and keep the rest as it is. In other words, the molecular viscosit is increased b the so called edd or turbulent viscosit. This assumption give us the possibilit to model the smallest vortexes b using correlations and approximations and onl resolve the larger eddies. It is also possible to use the higher viscosit to describe or characterie the dissipation of kinetic energ per unit mass of the turbulence into heat the higher the viscosit of the fluid, the higher the shearing and therefore we get higher mixing rates additional transport but although a larger dissipation of the kinetic energ into heat. Hence, we also could describe the theor vice versa: the higher the edd viscosit, the higher the turbulence of the flow field. Joseph Boussinesq related the Renold-Stresses ρu i u j to the mean values of the velocities and the kinetic energ of the turbulence k as: ρu ūi i u j = µ t + ū j 2 Ū δ ij 2 x j x i 3 3 ρδ ijk, 9.63 σ t 2 D 2 3 tr DI 2 3 ρik 2 σ t = 2µ t D 3 µ t tr DI 2 ρik, σ t = τ t 2 ρik In the equations above, we can see, that the underlined term is identical to the shear-rate tensor τ, if we set the bulk viscosit κ to ero; cf. equation 5.14 with

94 9.4. BOUSSINESQ EDDY VISCOSITY 87 the difference that we use the turbulent edd viscosit µ t instead of the molecular viscosit µ, hence we mark it with the subscript t τ t. In addition we can sa that the turbulent Renolds-Stress tensor σ t equals to the shear-rate tensor τ t and an additional term 2 3 ρδ ijk. This term is necessar to guarantee the proper trace of the Renolds-stress tensor σ t as mentioned b Feriger and Perić [2008] and Wilcox [1994]. One ma think about the term 2 3 ρδ ijk now. Where does it come from and wh is this term necessar? As mentioned b Feriger and Perić [2008] and Wilcox [1994], this term has to be added to get the proper trace of the Renolds-stress tensor. To understand the meaning of the additional term, we simpl have to take the trace of the Renolds-Stress tensor σ t and the shear-rate tensor τ t. For that, we need to know the definition of the kinetic energ of the turbulence: k = 1 2 u i u i = 1 2 u x u x + u u + u u New we need to take the trace of the Renolds-Stress tensor σ t It follows: ρu xu x ρu u x ρu u x tr σ t = tr ρu i u j = tr ρu xu ρu u ρu u, 9.67 ρu xu ρu u ρu u = ρu xu x + ρu u + ρu u = 2ρk The result that we get is the following: The trace of the Renolds-Stress tensor is twice the densit multiplied b the kinetic energ of the turbulence, 2ρk, and can be validated b substituting k b its definition 9.66: [ 1 ] 2ρk = 2ρ u 2 x u x + u u + u u = ρu xu x + ρu u + ρu u. tr σ t 9.69 We demonstrated, that the trace of the Renolds-Stress σ t tensor has to be equal to 2ρk. Therefore, the trace of the RHS of equation 9.65 has to be equal to 2ρk too. Otherwise the equation would not be correct in the mathematical point of view. Thus we get: tr σ t = tr τ t 23 ρik = 2ρk, 9.70

95 88 Turbulence Modeling 2 tr σ t = tr 2µ t D 3 µ t tr DI 2 3 ρik = 2ρk If we use the definition of the deformation rate tensor 5.11 with respect to the mean quantities and appl the transformation 8.9, it follows: tr σ t = tr { 1 2µ t [ 2 Ū + ]} Ū T 2 3 µ t Ū I Term 1 Term 2 T x ū x x ū x = tr µ t ū + ū ū ū 2 3 µ t ū x x + ū 0 + ū Term ū x x + ū 2 3 ρik, Term 3 + ū 0 ū x x + ū + ū 0 0 Term ρk ρk }{{ ρk } Term 3 Appling the dadic product rule 1.15 to term 1, add both matrices and multipl everthing b the edd viscosit, we end up with term 1 as: u µ x t x u µ x t u µ x t + µ t ux x + µ t u x + µ t u x u µ t x + µ t ux µ t u µ t u + µ t u + µ t u x + µ t ux + µ t u. + µ t u µ t u µ t u µ t u Due to the fact that we are onl interested in the main diagonal elements trace, we just consider these terms for now. The matrices of term 1, term 2 and term 3

96 9.4. BOUSSINESQ EDDY VISCOSITY 89 have to be summed up and the trace operator has to be applied. It follows: tr τ t 23 ρik [ ūx = 2µ t x + ū + ū ] 3 2 [ 3 µ ūx t x + ū + ū ] Term 1 Term 2 = ρk = 2ρk Term 3 The result of the trace operator to the Boussinesq hpothesis is 2ρk. Hence, the trace of the RHS and LHS of equation 9.63 is equal. If we would remove the term 2 3 ρδ ijk on the RHS of equation 9.63, the trace of the RHS would not be equal to the trace of the Renolds-Stress tensor σ t and hence, the Boussinesq edd viscosit assumption would be wrong because τ t is traceless, cf. chapter 6. Forums Discussion It is worth to mention that there were a lot of people asking about the term 2 3 ρδ ijk in public forums. Even I made wrong statements at the beginning in a wa that within the OpenFOAM R toolbox, this term is neglected. Finall, we cannot find this term in OpenFOAM R which is related to a ver simple correlation that is given below. For those who are interested in the discussion on cfd-online.com, ou can go to: Keep in mind that this thread can cause confusion because onl the last posts are correct and as Gerhard Holinger mentioned, the term is put into a modified pressure and is not neglected in OpenFOAM R. How this is working, is given on the next page.

97 90 Turbulence Modeling Analog to the Cauch Stress Tensor, Shear-Rate Tensor and Pressure Comparing the last derived equations with those of chapter 6, it is obvious that there are similarities. Analing equation 6.7 and 9.65, we can evaluate the same kind of behavior: σ Cauch Stress tensor = τ shear rate tensor traceless + pi, 9.74 pressure =trace σ t = τ t ρik, 9.75 Renolds Stress tensor RA shear rate tensor traceless add. term =trace A complete matrix = A dev deviatoric part traceless + A}{{ hd } hdro. part =trace If we compare the terms, we can observe that the term 2 3ρk seems to behave like a pressure. Using equation 9.42 and replacing the Renolds-Averaged Cauch stress tensor σ and the Renolds-Stress tensor σ t b their definitions 6.7 and 9.65, we can highlight the similarities better: ρ + ρ Ū Ū = τ + pi tū σ +ρg + τ t + 23 ρki σ t, 9.77 ρ tū + ρ Ū Ū = τ + τ t + ρg + pi + 23 ρki, 9.78 ρ tū + ρ Ū Ū = τ + τ t + ρg pi ρki = p I Introducing a modified pressure p = p + 2 3ρk and replacing the shear-rate tensors b their definitions for the shear-rate tensor τ, we mark the molecular viscosit µ b the subscript l, we get the following equation: ρ tū + ρ Ū Ū = [ 2µ l D 2 [ 2 + 2µ t D 3 µ t 3 µ l Ū ] I Ū ] I + ρg p I, 9.80

98 9.4. BOUSSINESQ EDDY VISCOSITY 91 ρ [ { tū + ρ Ū Ū = µ l 2D 2 Ū }] I 3 { + [µ t 2D 2 Ū }] I + ρg p I, { ρū+ ρū Ū = [µ l + µ t ] t 2D 2 3 Ū } I +ρg p I Introducing an effective viscosit µ eff that is simpl the sum of the molecular and turbulent edd viscosit: µ eff = µ l + µ t, 9.83 we can rewrite the Renolds-Averaged-Navier-Stokes equations, that include the effective viscosit and a modified pressure field p as: { ρū+ ρū Ū = µ eff t ρū+ ρū Ū = t 2D µ eff D 3 µ eff Ū I τ eff Ū } I +ρg p I, ρg p I After introducing the effective shear-rate tensor τ eff, we can simplif the equation to: t ρū + ρū Ū = τ eff p I + ρg It is clear that this equation is similar to equation The differences within the equations are, that we use a modified pressure p and a new viscosit µ eff field; it should be obvious that we have mean quantities here. After that, the onl unknown in that equation is the edd viscosit µ t. If we introduce an effective Cauch-Stress tensor σ eff, we are able to build the general form of the momentum equation. It follows: σ eff = τ eff p I, 9.87 t ρū + ρū Ū = σ eff + ρg Note: If we solve the incompressible Renolds-Averaged Navier-Stokes equations, we are not calculating the real pressure field p. Instead we have the modified pressure p. For most of the problems this is not a big deal and we do not have to

99 92 Turbulence Modeling consider this. Onl if we are using some modified equations in OpenFOAM R where we need the real pressure, we have to recalculate the real pressure field b subtracting the kinetic part. The reason for introducing the Boussinesq edd hpothesis and the advantages are given in the next section. 9.5 Edd Viscosit Approximation The Boussinesq theor allows us to eliminate the Renolds-Stresses with known quantities. However, we get new unknown quantities like the edd viscosit µ t and the kinetic energ of the turbulence k. Wilcox [1994] listed plent of theories and models that relates the edd viscosit to known quantities. Commonl the turbulent edd viscosit is characteried with the kinetic energ of the turbulence k and a characteristic length L. Furthermore, the kinetic energ of the turbulence can be related to a velocit q = k. This two values enable us to derive a correlation between the velocit q kinetic energ of the turbulence k, the characteristic length L and the edd viscosit. The assumption that was invented is: µ t C µ ρql The parameter C µ is a dimensionless constant. The challenge now is to relate the characteristic length L and the velocit q to known quantities. This is done b using turbulence models. 9.6 Algebraic Models At the beginning of computational fluid dnamics the power of personal and super computers were restricted and therefore it was necessar to have simple models that approximate the Renolds-Stresses ρu i u j. These models commonl use the introduced Boussinesq edd viscosit theor. The estimation of the edd viscosit µ t is done b using algebraic expressions. A few models are described in Wilcox [1994] chapter 3. Algebraic models can be used for simple flow patterns but hence the flow is getting complex imagine geometries in combustion, or even flow separation, these models will fail and produces non-phsical values for the edd viscosit.

100 9.7. TURBULENCE ENERGY EQUATION MODELS Turbulence Energ Equation Models The most common approximations for the Renolds-Stresses finall to calculate the characteristic length scale L and the kinetic energ of the turbulence k are called turbulence energ equation models. There are one-equation and two-equation models. Due to the fact that we need the values of the velocit q = k and the characteristic length L, it is logical to use two-equation models, where each equation models one parameter. Therefore, we focus onl on this kind of approximations for now. In general the velocit q is calculated using the kinetic energ of the turbulence k. To evaluate k we can make use of the alread know relation between the trace of the Renolds-Stress tensor σ t and k, cf To get the equation of the kinetic energ of the turbulence k, we simpl have to take the trace of the Renolds-Stress equation. How we get this equations are discussed in the following sections. 9.8 Incompressible Renolds-Stress Equation To derive the Renolds-Stress equation, we will use the Navier-Stokes equation 5.10 with bulk viscosit equals to ero, no source terms and incompressibilit dilatation term is ero. The start point for the derivation is: ρ u i t + ρ x j u j u i = x j [ { 1 ui 2µ + u }] j p x j x i x i To make life easier, we will split the convective term b using the product rule. Hence, we can remove one term due to the continuit equation non-conserved equation. We get: ρ u i u j u j u i = ρu j + ρu x j x j i x j continuit Replacing the convective term with the new form, put everthing to the LHS and introduce the Navier-Stokes operator N, we get: ρ u i t + ρu u i j x j x j [ { 1 ui 2µ + u }] j + p = N u i x j x i x i It is clear that the Navier-Stokes operator N has to be equal to ero and thus we

101 94 Turbulence Modeling can write: N u i = With the new information, we are able to derive the Renolds-Stress equation. In order to form this equation we multipl the Navier-Stokes operator b the fluctuation with respect to the different space directions: u in u j + u jn u i = The next step is to appl the expression 9.1 to the Navier-Stokes operator N u i and average the whole equation b using the Renolds time-averaging method 9.9. This leads to the following equation: u i N ū j + u j + u j N ū i + u i = This equation has to be evaluated to get the Renolds-Stress equation. The derivation itself is not a big deal but we have to have the feeling for different behaviors of the terms and hence, we need to be familiar with the mathematics. In the following, we will give a brief summar of the operations and relations and present the Renolds-Stress equation without an derivation. The full derivation of this tensor equation is given in the appendix in section Operations and relations For the derivation of the Renolds-Stress equation we need to build the following tensor equation with formula 9.95: u xn ū x + u x + u N ū + u + u xn ū + u + u N ū x + u x +u xn ū + u + u N ū + u + u N ū x + u x + u N ū + u +u N ū + u + u N ū x + u x + u N ū + u + u N ū + u +u N ū x + u x + u xn ū + u + u N ū + u + u xn ū x + u x +u N ū + u + u xn ū + u = 0. For the derivation we further use the following relations, rules and tricks: Renolds time-averaged terms that are linear in the fluctuation are ero, The derivative u i x i = 0,

102 9.9. THE INCOMPRESSIBLE KINETIC ENERGY EQUATION 95 Product rule 1.2, Adding and subtracting terms to be able to use the product rule; gx = gx + fx fx. If we use these assumptions, we can derive the Renolds-Stress equation. result of the derivation procedure is a more or less complex equation. Hence, the Renolds-Stress equation is given as: σ tji t + ū k σ tij u k = σ tjk ū i x k σ tik ū j x k + x k ν σ t ij x k + C ijk The + ɛ ij Π ij After knowing the Renolds-Stress equation, we are able to derive the equation for the kinetic energ of the turbulence k. The Renolds-Stress equation also gives insight into the nature of the turbulent stresses and can be used to understand the turbulence in more detail or it can be used for further investigations in deriving more accurate turbulence models. 9.9 The Incompressible Kinetic Energ Equation The derivation of the kinetic energ equation of the turbulence per unit mass for incompressible flows is simple after knowing the Renolds-Stress equation due to the fact of the relation given b equation In order to get the equation, we have to take the trace of equation It follows: { } σtji σ tij tr + ū k t u k { ū i ū j = tr σ tjk σ tik + x k x k x k ν σ t ij x k After appling the trace operator to each term we get: For the time derivation we get: { } σtji tr t = σ t xx t + σ t t + σ t t = ρ u xu x t + C ijk = ρ t u xu x + u u + u u = 2ρ k t k + ɛ ij Π ij } ρ u u t ρ u u t. 9.98

103 96 Turbulence Modeling If we appl the trace operator to the convective term, we get: { } σ tij σ txx tr ū k = ū k u k u k = ρū k u xu x u k + ū k σ t u k + ū k σ t u k ρū k u u u k ρū k u u u k = ρū k u k u xu x + u u + u u = 2ρū k k u k The first and second term of equation 9.96 lead to: { } ū i ū j ū i ū i tr σ tjk σ tik = σ tik σ tik = 2ρu ū i i x k x k x k x u k k x k The first part of the third term results in: { tr ν σ } t ij = ν σ t xx x k x k x k x k = x k = x k = 2 x k + x k ρν u xu x x k µ ν σ t x k x k The second part of the third term, C ijk, results in: + x k ρν u xu x x k ν σ t x k x k ρν u xu x x k u x xu x + u u + u u k µ k x k { tr ρu i x u j u k + [ ] } p k x u j δ ik + p u i δ jk = ρu j k x u i u i + 2 p j x u j. j The evaluation of the second term that includes the pressure can be done in an eas wa. It is simpl twice the trace of one of the terms. The first term is a third rank tensor T 3 and the trace results in the underlined term on the RHS. This can be demonstrated b analing the first entries of the third rank tensor: tr ρu xu j u k ρu xu xu x ρu xu xu ρu xu xu = tr ρu xu u x ρu xu u ρu xu u ρu xu u x ρu xu u ρu xu u = ρu xu i u i

104 9.9. THE INCOMPRESSIBLE KINETIC ENERGY EQUATION 97 In a similar wa to C ijk, the term ɛ ij can be manipulated. Thus, we get: tr { 2µ u i x k u j x k } = 2µ u i x k u i x k The last term of equation 9.96, Π ij, is ero due to the fact that u i x i = 0: tr { p [ u i x j + u j x i ] } [ = p u x x + ] u x x [ ] + p u + u [ + p u + ] u = If we sum up all terms, we get the kinetic energ equation of the turbulence, k, for incompressible fluids: 2ρ k t 2ρū k k = 2ρu ū i i u u k 2 k x k x k µ k x k + x k ρu j u i u i + 2 x k p u i + 2µ u i x k u i x k Finall, we divide the whole equation b 2 to get the final form of the incompressible kinetic energ equation: ρ k t + ρū k k = ρu ū i i u u k + µ k k x }{{ k x } k x k P k ρ x k 2 u j u i u i p x u j µ u i u i j x k x k turbulent diffusion dissipation ɛ If we merge the terms that include the turbulent diffusion, use the term P k, that denotes the production rate of the kinetic energ and the acronm ɛ, we end up with the common kinetic energ equation as: ρ k t + ρū k k = u k x k µ k + P k [ ρ ] x k x k 2 u j u i u i + p u j ɛ The production rate and turbulent diffusion term has to be modeled. For the tur-

105 98 Turbulence Modeling bulent diffusion we use the assumption that the diffusion is based on the gradients: [ ρ 2 u j u i u i + p u j ] µ t Pr t k x j Here, Pr t denotes the turbulent Prandtl number and is assumed to be one. In the literature we it is also common to denote the turbulent Prandtl number b σ k. Due to the fact that we use sigma to describe an kind of stress tensor, we avoid the usage of another sigma quantit here. The production rate is modeled with the assumption given b equation 9.63 but with the difference, that we do not need the term 2ρk; Recall: This term was just added to equilibrate both sides. Hence, the production rate term is given b: P k = ρu ū i i u j µ t x j ūi + ū j ūi x j x i x j The dissipation ɛ, that describe the transfer of the turbulence into internal energ a better description given in the next section, is coupled to a characteristic length scale L. Recall: After we derived the RANS equations, we figured out that we need to calculate the Renolds-Stress tensor σ t. To get this value we introduced the Boussinesq edd viscosit hpothesis and related the edd viscosit µ t to a characteristic length scale L and a velocit q. Up to now, we eliminated one unknown q = k b using the kinetic energ k but we also introduced a new unknown quantit, the dissipation ɛ. Thus, we still have two unknown, the length scale L and the dissipation ɛ. The good thing is, both quantities can be related The Relation between ɛ and L The most common equation that is used to estimate the length scale L is based on the observation that the dissipation phenomena can also be observed in the energ transport and thus in its equation. In fluid flows which are in a so called turbulent equilibrium, it is possible to derive a relation between the kinetic energ k, the length scale L and the dissipation ɛ: ɛ k 3 2 L

106 9.11. THE EQUATION FOR THE DISSIPATION RATE ɛ 99 The idea behind this relation is the so called energ cascade for high turbulent flow fields high Renolds numbers. The concept can be described as follows: The kinetic energ of the turbulence is transformed from big scale eddies to small scale eddies. If we reach the smallest scale this vortexes are named Kolmogorov vortexes, the viscous effect will transfer the energ of motion into internal energ. This phenomena is called dissipation The Equation for the Dissipation Rate ɛ To calculate the length scale L and close the equation for the turbulent energ k, we need the equation for the dissipation ɛ. This equation can be derived b using the Navier-Stokes equation like we did for the Renolds-Stress equation but due to the fact that most terms on the RHS have to be modeled, we should describe this equation more like a model than an exact equation. Hence, the complete derivation is not shown. In general we are using the following equation for the dissipation ɛ: ρ ɛ t + ρū ɛ ɛ j = C ɛ1 P k x j k ρc ɛ 2 ɛ 2 k + x j µt σ ɛ ɛ x j As we can see, the whole right side is more like a plaground of parameters and assumptions than a real fundamental equation. But the equation allows us to estimate the dissipation ɛ. The quantit can be used for the kinetic energ equation and allows us to estimate the length scale L and hence, we are able to approximate the the edd viscosit µ t. Now we can rewrite the Boussinesq edd hpothesis 9.89 with the new quantities: k 3 µ t = ρc µ kl = ρcµ k ɛ k 2 = ρc µ ɛ The model parameters of the equations above are: C µ = 0.09, C ɛ1 = 1.44, C ɛ2 = 1.92, σ ɛ = 1.3.

107 100 Turbulence Modeling 9.12 Coupling of the Parameters As we could see in the last sections, the turbulence modeling is a complex topic. The easiest equations for the turbulence modeling were derived. Furthermore, we observed that all parameters are coupled. The kinetic energ of the turbulence k, the length scale L, the dissipation ɛ and the edd viscosit µ t. There are a lot of more considerations that have to be taken into account if turbulence modeling is used. Just think about the turbulence behavior close to the walls compared to the far field. Another example would be the turbulence modeling of flow separation. The section about the turbulence model gave us a feeling that the topic about turbulent flows are extreme complex. A lot of research was done and till toda the turbulence has still to be modeled and can onl be applied and resolved with all details for a couple of problems. In the literature we will find different equations that give reasonable results for a special kind of problems. Good references for further investigations into the turbulence modeling are the books of Feriger and Perić [2008], Bird et al. [1960] and Wilcox [1994] Turbulence Modeling for Compressible Fluids As alread discussed during the Renolds averaging procedure for the incompressible mass conservation equation, the varing densit has to be taken into account during for compressible fluids. Therefore, we get: ρ = ρ + ρ This lead to more unknown terms that will make the problem even more complex; compare the alread derived Renolds time-averaged compressible mass conservation equation 9.24 which is given again: ρ t + ρū i + ρ x u i = i Here we need approximations for the correlation between ρ and u i, that are vanishing for incompressible fluids. If we go further and think about the momentum equations, we can imagine, that things get even worse. To get rid of the additional correlation between ρ and u i, we introduce a mathematical method suggested b Favre. This concept is based on mathematics

108 9.13. TURBULENCE MODELING FOR COMPRESSIBLE FLUIDS 101 and therefore not phsical correct. What we do is simple. We introduce a massaveraged velocit field ũ i, that is defined as: ũ i = t+t lim ρt, xu i t, xdτ ρ T t Here, ρ denotes the Renolds time-averaged densit and the tilde above the velocit u i marks the quantit to be Favre averaged instead of Renolds averaged. In terms of the Renolds time-averaging procedure we are allowed to sa: ρũ i = ρu i To show what happens here, we will expand the RHS: ρũ i = ρ + ρ ū i + u i = ρū i + ρu i + ρ ū i + ρ u i = ρū i + ρ u i If we use this expression for equation 9.115, we end up with: ρ t + ρũ i = x i This equation looks just similar to the laminar mass conservation or the Renolds time-averaged equation. Wilcox [1994] explained this kind of averaging as follows: What we have done is treat the momentum per unit volume, ρu i, as the depended variable rather than the velocit. phsical point of view,... This is a sensible thing to do from a If ou are interested into that kind of averaging, a lot of information can be found in Wilcox [1994] and Bird et al. [1960]. Note: The ke argument to use the Favre averaging procedure is to simplif the averaging procedure and get rid of additional correlations that have to be modeled. Hence, we end up with the same set of equations for incompressible turbulent flow fields but now we use the Favre-weighted quantities.

109 102 Turbulence Modeling

110 Chapter 10 Calculation of the Shear-Rate Tensor in OpenFOAM R In this chapter we will discuss the implementation of the calculation of the shearrate tensor τ, τ t or τ t ; real, Renolds-averaged or Favre-averaged quantities. In OpenFOAM R we calculate the shear-rate tensor b calling the functions divdevreff or divdevrhoreff. The following discussion is based on the OpenFOAM R version Be prepared that the code can look different for other versions. If ou are asking ourself for that reason, ou have to understand the concept of hacking. For those who are interested ou can read the book of Erickson [2014] The Inco. Shear-Rate Tensor, divdevreff For incompressible fluids, the momentum equation that is going to be constructed in the UEqn.H file looks equal to the following code snippet from pimplefoam: 1 tmp < fvvectormatrix > UEqn 2 3 fvm :: ddt U I 4 + fvm :: div phi, U II 5 + MRF. DDt U III 6 + turbulence -> divdevreff U IV 7 == 8 fvoptions U V 9 ; Listing 10.1: $FOAM SOLVERS/incompressible/pimpleFoam/UEqn.H 103

111 104 Calculation of the Shear-Rate Tensor in OpenFOAM R Lets anale the code. First of all we observe different terms. The first one is the time derivation I. The second term is the convective term II. After that, some additional correction term due to MRF III is added. The next one is the the shear-rate tensor IV and finall we have a term V that handles additional sources within the fvoptions dictionar. For now we will focus on the term IV turbulence->divdevreffu. First of all, we will discuss the meaning of the name divdevreff: τ = σ dev 10.1 The divergence of the shear-rate tensor is equal to the divergence of the deviatoric part of the stress tensor σ. The R comes from the Renolds-Average approach. An additional Rho defines if we are using a densit based or non densit based solver. Finall we are interested in the effective transport which includes laminar and turbulent transport phenomena. Therefore, we get the name divdevreff for incompressible and divdevrhoreff for compressible fluids. The anale of the function is ver simple and can be checked with the code source guide named Doxgen. The object named turbulence, which is based on the general turbulencemodel class and further derived from the incompressibleturbulencemodel class, will call the function divdevreffu. The corresponding function that is called and implemented is: 1 tmp < fvvectormatrix > laminar :: divdevreff volvectorfield & U const 2 { 3 return fvm :: laplacian nueff, U 6 - fvc :: div nueff *dev T fvc :: grad U 7 ; 8 } Listing 10.2:.../turbulenceModel/incompressible/RAS/laminar/laminar.C First, we can see that we have the kinematic viscosit included, which indicates incompressible equations. The second thing that can be observed is the fact, that we calculate one part implicit fvm and another part explicit fvc. Furthermore, a new function named dev is called.

112 10.1. THE INCO. SHEAR-RATE TENSOR, DIVDEVREFF 105 The new function dev is implemented as: 1 template < class Cmpt > 2 inline Tensor < Cmpt > dev const Tensor < Cmpt >& t 3 { 4 return t - SphericalTensor < Cmpt >:: onethirdsi * tr t; 5 } Listing 10.3: $FOAM SRC/OpenFOAM/primitives/Tensor/TensorI.H Analing the C++ Functions Starting with the function dev, we can see that the function simpl calculates the deviatoric part of the matrix, cf Therefore, we can write: A dev = A A hd = A 1 trai In addition, the function dev needs an argument. The argument is the transposed matrix that we get, if we build the dadic product of the two vectors, Nabla and the velocit vector U. Thus, we can write: A = [ gradū] T = [ Ū ] T If we express the C++ code with the expressions above, we can rewrite the code into a mathematic form. Hence, the first term of the divdevreff function can be written as: fvm :: laplacianν eff, Ū = ν eff Ū, 10.4 the second term is equal to: fvc :: divν eff devtfvc :: gradū = ν eff dev ŪT, 10.5 and the dev-function is similar to: dev ŪT = ŪT 1 3 tr ŪT I Finall, we are able to put everthing together and end up with: τ eff = [ ν eff Ū ν eff ŪT 1 ] 3 tr ŪT I. 10.7

113 106 Calculation of the Shear-Rate Tensor in OpenFOAM R After we push the first term into the second one: τ eff = ν eff Ū + ν eff ŪT 1 3 ν eff tr ŪT I, 10.8 and rewrite the last term b using the relation 8.9, we end up with: τ eff = ν eff Ū + ν eff ŪT 1 3 ν eff }{{ Ū I } continuit The last term is ero due to the continuit equation. Therefore, the effective shearrate tensor τ eff for incompressible fluids is calculated as: τ eff = ν eff Ū + ν eff ŪT τ eff [ 1 = 2ν eff 2 { }] Ū + ŪT deformation rate tensor D As we demonstrated now, the calculation of the incompressible shear-rate tensor is correct implemented into the version because the derived equation is similar to equation The sign difference of the term corresponds to the position at the LHS in OpenFOAM R whereas in equation 5.35 the shear-rate tensor stands on the RHS. Stabilit During the anale of the function, we figured out that the term 1 3 ν eff tr U is kept. The reason for that is based on numerics. The term simpl stabilies the calculation because the continuit equation is never 100% ero. This is based on the discretiation, interpolation and accurac of the machine. In newer OpenFOAM R versions like 4.x, the incompressible solvers will call dev2 that is normall used for compressible flows. The difference of dev2 and dev will be shown later and is simple a difference of the factor two at some position of the code. However, due to the continuit equation, we can cancel the additional term again and hence, the equation is also valid. If we are using dev2 instead of dev for incompressible solvers will lead to an even better stabiliation and convergence rate. This is the reason wh it was introduced in OpenFOAM R

114 10.2. THE COMPR. SHEAR-RATE TENSOR, DIVDEVRHOREFF The Compr. Shear-Rate Tensor, divdevrhoreff If we focus on compressible fluids, the shear-rate tensor has a different formulation. The constructed momentum equation looks similar to the one we had for incompressible fluids but now we have the densit included and we call a new function named divdevrhoreff. The code snippet is based on rhopimplefoam. 1 tmp < fvvectormatrix > UEqn 2 3 fvm :: ddt rho, U I 4 + fvm :: div phi, U II 5 + MRF. DDt rho, U III 6 + turbulence -> divdevrhoreff U IV 7 == 8 fvoptions rho, U V 9 ; Listing 10.4: $FOAM SOLVERS/compressible/rhoPimpleFoam/UEqn.H As before, we have the time derivation I, the convective II, some additional correction due to MRF III, the shear-rate tensor IV and the term V that handles additional sources within the fvoptions dictionar. For the analsis of the function, we proceed like before. First we investigate into the call of the function turbulence->divdevrhoreffu. Note: The keword Rho is now included in the name of the function. This indicates that we calculate the shear-rate tensor based on the theor for compressible fluids. Thus, the dilatation term is included due to expansion and compression phenomena which can be related to the non-constant densit. The shear-rate tensor that is calculated for a compressible fluid is given below. 1 tmp < fvvectormatrix > laminar :: divdevrhoreff volvectorfield & U const 2 { 3 return fvm :: laplacian mueff, U 6 - fvc :: div mueff * dev2 T fvc :: grad U 7 ; 8 } Listing 10.5:.../turbulenceModel/compressible/RAS/laminar/laminar.C The function is similar to dev but now we have the molecular instead of the kinematic viscosit and call a new function named dev2. The code of the new

115 108 Calculation of the Shear-Rate Tensor in OpenFOAM R function is presented below. It is somehow calculating the deviatoric part of a tensor but subtraction twice the hdrostatic part instead of once. The reason for that is obvious after we anale the code snippet. 1 template < class Cmpt > 2 inline Tensor < Cmpt > dev2 const Tensor < Cmpt >& t 3 { 4 return t - SphericalTensor < Cmpt >:: twothirdsi * tr t; 5 } Listing 10.6: OpenFOAM/primitives/Tensor/TensorI.H Analing the C++ Functions The argument that is return b the function dev2 represents equation 1.31 with the alread mentioned difference that we subtract the hdrostatic part twice; twothirdsi: A dev = A 2A hd = A 2 trai The argument of the function dev2 is equal to the one we had in the incompressible case. Hence, it can be evaluated b Rewriting the C++ code into the different equations, we are able to rewrite the first term as: fvm :: laplacianµ eff, Ũ = µ eff Ũ, the second term to: fvc :: divµ eff dev2tfvc :: µ gradũ = eff dev2 ŨT and the dev2-function like:, dev2 ŨT = ŨT 2 3 tr ŨT I After combining these terms, it follows: [ τ eff = µ eff Ũ µ eff ŨT 2 ] 3 tr ŨT I

116 10.3. INFLUENCE OF TURBULENCE MODELS 109 After we pushed the divergence operator out, we get: [ τ eff = µ eff Ũ + µ eff ŨT 2 ] 3 tr ŨT I B eliminating the brackets inside, it follows: τ eff = µ eff Ũ + µ eff ŨT 2 3 µ eff tr ŨT I, and finall, we get the known shear-rate tensor b using equation 8.9 as: τ eff = µ eff Ũ + µ eff ŨT 23 µ eff ŨI effective shear rate tensor τ eff To get a more familiar equation, we do some simple mathematics and end up with: [ 1 }] τ eff = 2µ eff { Ũ + ŨT µ eff ŨI deformation rate tensor D As we demonstrated, the calculation of the compressible shear-rate tensor is implemented in OpenFOAM R correctl. The equation above is equal to the averaged shear-rate tensor τ in equation 9.85 with the difference that we use Favre averaged quantities here. As before, the difference in the sign is based on to the fact that the term stands on the LHS in OpenFOAM R Influence of Turbulence Models If we use a compressible based solver in OpenFOAM R and simulate a laminar flow pattern, the momentum equation will not change based on the fact that it is hard coded. The question now is, what happens if we do so not use a turbulence model? As we saw in the chapters above, the equations for full resolved eddies, Renolds- Averaged or Favre-Averaged flow fields are identical. The onl difference is related to the viscosit. Hence, if we do not use a turbulence model, the contribution of the edd viscosit µ t is ero. It follows: µ eff = µ l + µ t

117 110 Calculation of the Shear-Rate Tensor in OpenFOAMr

118 Chapter 11 SIMPLE, PISO and PIMPLE algorithm Solving the Navier-Stokes equations requires numerical techniques for coupling the pressure and momentum quantities. This is done b the common SIMPLE, PISO and PIMPLE algorithms. Wh we need these numerical methods can be understood after we introduce the difficulties that come into hand when we solve the equations. Further information about these algorithms can be found in Feriger and Perić [2008] and Moukalled et al. [2015]. Let us consider the general momentum equation 2.26 and appl the incompressibilit character the densit is constant and can be taken out of the derivatives, we get: ρ U t = ρ U U τ p + ρg Note: In the above equation we did not introduce the shear-rate tensor. Thats wh the sign is negative. Now we divide the whole equation b the densit ρ. Hence, τ has to be expressed b equation 5.35 and it follows: U t = U U τ inco p ρ + g Based on the fact that the densit is constant, there is no need to solve an energ equation and we onl need to solve the momentum equation. As we can see, there are four unknown quantities, the pressure p and the three velocit components 111

119 112 SIMPLE, PISO and PIMPLE algorithm denoted b U. The problem is, that we have four unknown and three equations momentum in x, and. The remaining equation that we can use it the mass conservation equation 2.13 but this equation even does not have the pressure included and therefore we need some special techniques to solve the momentum equation. This is also known as the pressure-momentum coupling problem. The idea, to get rid of the problem, is to use the mass conservation equation somehow. What we finall do, is to appl the divergence operator onto the momentum equation. After doing a semi-discretiation, that means, we onl discretied the time derivative while the space derivatives are kept in partial differential form, we can use the mass conservation equation to eliminate terms and end up with the well known Poisson equation for the pressure p. Now we have an equation for the momentum and the pressure. In general these equations are solved sequential. That means, that we solve for U x while we keep all other variables constant. In other words, we first solve the equation for U x, then for U, then for U and then for p. The strateg now is to find a pressure and momentum field that fulfill the mass conservation and of course the case conditions at some defined time. This is achieved within the pressure-momentum coupling algorithms PISO, SIMPLE and PIMPLE. As a simple rule not valid all the time, we can sa: SIMPLE:= Semi-Implicit-Method-Of-Pressure-Linked-Equations. In OpenFOAM R we are using this algorithm for stead-state anales. PISO:= Pressure-Implicit-Split-Operator. In OpenFOAM R we are using this algorithm for transient calculation. The calculation is limited in the time step based on the Courant number. PIMPLE:= Merged PISO SIMPLE. This algorithm combines both algorithms and allows us to use bigger timesteps Co >> 1. Note: There is a famil of different algorithms available. For example, the SIMPLE algorithm is not consistent. That means, that during the derivation of the pressure equation, we neglect one term. Therefore, different investigations were done to calculate the missing term and are known under the terms of SIMPLER, SIMPLEM, SIMPLEC and so on. A ver good overview is given in Moukalled et al. [2015].

120 11.1. THE SIMPLE ALGORITHM IN OPENFOAM R The SIMPLE algorithm in OpenFOAM R If we use an kind of SIMPLE based solver in OpenFOAM R, we do not have a time derivation. A time derivation is normall a natural limiter for the solution. That means, for a special time interval t, the solution can onl go on b this time step and not further. Based on the fact that we do not have the time derivation within that algorithm, we are onl interested in the stead-state behavior and based on the missing natural limiter t and the fact that the SIMPLE algorithm is not consistent missing term, we need to under-relax the equations to achieve stabilit. Otherwise, the solver just blow up and gives a floating point exception dividing b ero. Furthermore, the time step t should be alwas set to 1. Doing that, the time will indicate the number of iterations that we did within the SIMPLE loop. Changing the time step to other values will not influence the solution. It simpl let us reach the pseudo end time faster or not. In other word, we do more or less iterations. Of course, changing t can affect the results but onl if we will do not reach the stead state solution. For the SIMPLE algorithm it is ver important to estimate the relaxation factors for the fields and equations for good stabilit and fast convergence rate SIMPLEC in OpenFOAM R In the release version 3.0.0, the SIMPLEC algorithm can be used in all SIMPLE and PIMPLE operating algorithms. For that purpose, we have to add some special keword to the SIMPLE or PIMPLE control dictionar in the fvsolutions file. Activating the SIMPLEC method can be done b adding the following keword: 1 SIMPLE 2 { 3 consistent true ; 4 } 5 6 PIMPLE 7 { 8 consistent true ; 9 } Listing 11.1: SIMPLE operating in SIMPLEC mode

121 114 SIMPLE, PISO and PIMPLE algorithm As alread mentioned in the previous section, the SIMPLEC algorithm include the missing pressure term. The added character C stands for consistenc. Using the SIMPLEC method will require more iterations for each single segregated calculation step but the convergence rate will increased. The release notes report a speed-up of three times. Furthermore, larger values for the under-relaxation factors can be choosen. Note: The consistenc keword can be added to the PIMPLE dictionar too, but will onl affect the algorithm, if we operate in the PIMPLE mode. Otherwise, this keword will not affect the numerical procedure. How to use the PIMPLE algorithm correct, will be discussed later The PISO algorithm in OpenFOAM R The two main differences to the SIMPLE algorithm are the included time derivation term and the consistenc of the pressure-velocit coupling equation. Based on this two additional criteria, we do not need to under-relax the fields and equations but need to fulfill a stabilit criterion. Based on the simulation tpe, we have to make sure that the so called Courant number is not larger than one. The Courant number can be visualied as follows: If the dimensionless number is smaller than one, the information from one cell can onl reach the next neighbor cell within one time step. Otherwise, the information can reach a second or third neighbor cell which is not allowed based on some explicit aspects. Therefore, the Courant number has to be smaller than one. In general, a small value has to be considered at the beginning of a simulation, which can then be increased to some case depended value. Co = U t x The Courant number depends on the local cell velocit U, the time step t and the distant between the cells x. In OpenFOAM R, the calculation is based on the cell volume and not on the distance x. Based on formula 11.3, we can derive the following aspects: The higher the local cell velocit U, the larger the Courant number, The larger the time step t, the larger the Courant number, The smaller the distance x, the larger the Courant number.

122 11.3. THE PIMPLE ALGORITHM IN OPENFOAM R 115 The main aspect here is, that if we refine the mesh, increase the velocit or the time step, the Courant number will increase. To fulfill the criteria in equation 11.3, the time step has to be adjusted based on the mesh sie and the velocit. Note: The criteria has to be fulfilled for each cell. That means, that one bad cell can limit the whole simulation The PIMPLE algorithm in OpenFOAM R The PIMPLE algorithm is one of the most used one if we have transient problems because it combines the PISO and SIMPLE SIMPLEC one. The advantage is, that we can use larger Courant numbers Co >> 1 and therefore, the time step can be increased drasticall. The principal of the algorithm is as follows: Within one time step, we search a stead-state solution with under-relaxation. After we found the solution, we go on in time. For this, we need the so called outer correction loops, to ensure that explicit parts of the equations are converged. After we reach a defined tolerance criterion within the stead-state calculation, we leave the outer correction loop and move on in time. This is done till we reach the end time of the simulation. Note: The PIMPLE algorithm in OpenFOAM R can also work in PISO mode, if we set the noutercorrectors to one. The value of ero just ignores the whole pimple loop no calculation. This can be checked, if we start an PIMPLE solver. The output should be as follow: 1 Create mesh for time = PIMPLE : Operating solver in PISO mode Listing 11.2: The output of the pimple algorithm

123 116 SIMPLE, PISO and PIMPLE algorithm 11.4 The correct usage of the PIMPLE algorithm The usage of the PIMPLE algorithm is explained and discussed within this section. First of all, the settings that can be set for controlling the algorithm has to be written into the fvsolution dictionar. Here, we need to define the algorithm control dictionar named PIMPLE. 1 PIMPLE 2 { 3 // - Settings that we can made 4 } Listing 11.3: The control dictionar within the fvsolution file The keword has to be added to the fvsolution file, otherwise the solver will throw out an error. Nevertheless, it is sufficient just to create an empt dictionar. If we do so, the code will use default values based on the constructor of the class. 1 // * * * * * * * * * * * * * Constructors * * * * * * * * * * * // 2 3 Foam :: pimplecontrol :: pimplecontrol 4 5 fvmesh & mesh, 6 const word & dictname 7 8 : 9 solutioncontrol mesh, dictname, 10 ncorrpimple_ 0, 11 ncorrpiso_ 0, 12 corrpiso_ 0, 13 turbonfinaliteronl_ true, 14 converged_ false 15 { 16 read ; Listing 11.4: The constructor of the pimplecontrol class As we can see, the constructor will initialie all values with ero first, and set two booleans. One to true and the other one to false. After that, we go into the read function. Here, we will read the PIMPLE dictionar in the fvsolution file. If there is no entr to read, the default values are set. For the noutercorrectors a value of one is used. The same is valid for the ncorrectors. Finall the turbonfinaliteronl is set to false. If this switch is turned on, we solve the

124 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM 117 turbulence equation within each outer loop, otherwise we onl solve it at the last iteration, after we leave the outer loop to go on in time. 1 // * * * * * * * * * Protected Member Functions * * * * * * * * // 2 3 void Foam :: pimplecontrol :: read 4 { 5 solutioncontrol :: read false ; 6 7 // Read solution controls 8 const dictionar & pimpledict = dict ; 9 ncorrpimple_ = pimpledict. lookupordefault < label > " noutercorrectors ", ; 14 ncorrpiso_ = pimpledict. lookupordefault < label >" ncorrectors ", 1; 15 turbonfinaliteronl_ = 16 pimpledict. lookupordefault < Switch > " turbonfinaliteronl ", 19 true 20 ; 21 } Listing 11.5: The read function of the pimplecontrol class In addition, we see that the noutercorrectors are related to the PIMPLE and the ncorrectors inner loops to the PISO algorithm. The read function will also call another readargument function from the solutioncontrol class. Here, we initialie other essential algorithm control parameters like the nnonorthogonalcorrectors, momentumpredictor, transonic, consistent and residualcontrol not shown in the code. 1 // * * * * * * * * * Protected Member Functions * * * * * * * * // 2 3 void Foam :: solutioncontrol :: read const bool abstolonl 4 { 5 const dictionar & solutiondict = this - > dict ; 6 7 // Read solution controls 8 nnonorthcorr_ = 9 solutiondict. lookupordefault < label > 10

125 118 SIMPLE, PISO and PIMPLE algorithm 11 " nnonorthogonalcorrectors ", ; 14 momentumpredictor_ = 15 solutiondict. lookupordefault " momentumpredictor ", true ; 16 transonic_ = solutiondict. lookupordefault " transonic ", false ; 17 consistent_ = solutiondict. lookupordefault " consistent ", false ; // Read residual information 20 const dictionar residualdict solutiondict. suboremptdict " residualcontrol " 23 ; // Residual controls not shown Listing 11.6: The read function of the solutioncontrol class To sum up. If we use the PIMPLE algorithm without an specific keword, we will have the following set-up: noutercorrectors ncorrpimple is set to 1, ncorrectors ncorrpiso is set to 1, nnonorthogonalcorrectors corrpiso is set to 0, turbonfinaliteronl is set to false, momentumpredictor is set to true, transonic is set to false, consistent is set to false, false means using SIMPLE; true means using SIMPLEC No residual control information is set. For the further anale, a simple non-stead test case is provided and the ke-ideas of the PIMPLE algorithm are demonstrated.

126 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM The test case To demonstrate the behavior of different settings within the fvsolution file, a simple 2D transient pipe flow will be considered; this is not the best case because the big advantage of the PIMPLE method comes with complex geometries. The pipe is contracted in the middle in order to accelerate the fluid and to create some vortex after that; compare figure The kinematic viscosit ν is set to 1e 5 and the extrusion of the 2D mesh in -direction is 0.01m. The OpenFOAM R version that is used is 4.x and the case is available on x = 0.45m x = 0.05m x = 0.2m noslip noslip = 0.025m = 0.1m inlet U = 0.2 m s outlet = 0.05m noslip noslip Figure 11.1: The 2D pipe flow domain for the anale of the PIMPLE algorithm. Another example and further explanations about the equations that are solved can be found in Holmann [2014] and an even more detailed introduction into the method and how the pressure-momentum coupling is done in OpenFOAM R is given in Moukalled et al. [2015] First considerations In the following, we are using the above domain with the mentioned boundar conditions to demonstrate the usage of the PIMPLE algorithm. First of all, we investigate into some basic behavior of the problem. With the usage of the Bernoulli equation, we can estimate the maximum time step for the simulation. The area at the inlet is A = 0.1m 0.01m = 0.001m 2.

127 120 SIMPLE, PISO and PIMPLE algorithm Therefore, we get a volumetric flux φ = UA = 0.2 m s 0.001m2 = m3 s. This flux has to cross the small section in the middle of the domain and thus, the velocit has to increase, while the pressure will drop. A cross = 0.025m 0.01m = m 2. The cross section has an area of Hence, the velocit in the cross section became around U cross = 0.8 m s. Based on the fact that the flow will be contracted in the cross section, the velocit will further rise. So let us assume that we will get the maximum velocit of 1.2 m s somewhere in the mesh for the first guess. The resulting time step, with a cell distance of x = 0.002m, can then be evaluated with equation 11.3: t = s = s. 1.2 The generated mesh is a pure hexaeder mesh and thus we do not need an orthogonal corrections. First, we will use the pisofoam solver and the evaluated time step as a reference for the residuals and velocit contour plot. After that, we will use the pimplefoam solver and appl different kewords and settings. To make the simple problem more complex, we use the Gauss linear discretiation scheme, that tends to produce non phsical results, if the stabilit criterion is not strictl fulfilled Run the case with the PISO algorithm The first step is to generate a reference case. solver. This is done using the pisofoam Due to the fact, that we solve the flow without a turbulence model, we will naturall get some vortexes which result in high transient behavior. This will make the sstem more stiff and harder to solve. Furthermore, we can expect, that the residual plots for the time steps look similar for each case, based on the transient character. At last, based on the high transient behavior, we will see some ver interesting numerical phenomenon at the end of the discussion. After the solver finished its work, we get the simulation results shown in figure 11.2 and Here, we used a fixed time step not adjustable. That means, based on the fact that the velocit will change during the simulation, the Courant number will change too, cf During the simulation a maximum Courant number of 1.4 and a maximum magnitude of the velocit of around 1.3 m s are achieved. This is the first indication,

128 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM 121 that our time step approximation works fine for this case. Although we reach higher Courant number than one, the simulation is still stable. It should be clear that the time step approximation based on equation 11.3 can onl be used for ver simple geometries; moreover it is common to adjust the time step based on the Courant number than using a fixed t. Another thing that we can observe in front of the contraction area are velocit stripes. This is based on the Gauss linear scheme and indicate that we are close to the instabilit limit and is a common habit of that scheme. 1e+00 1e-01 PISO algorithm p Ux U Residuals [-] 1e-02 1e-03 1e-04 1e Time [s] Figure 11.2: Residual plot for the pisofoam solver; fixed t = s. Figure 11.3: Velocit contours generated with pisofoam and a fixed t = s.

129 122 SIMPLE, PISO and PIMPLE algorithm PIMPLE working as PISO Now we are using the same case without an changes and run the pimplefoam solver on it. Based on the fact that the PIMPLE entr within the fvsolution is empt, OpenFOAM R will use default values and hence, the pimplefoam solver is equal to pisofoam. That can be proofed b checking the residuals and contour plots of this calculation and the last one. We see that both solutions are equal. 1e+00 1e-01 PIMPLE algorithm p Ux U Residuals [-] 1e-02 1e-03 1e-04 1e Time [s] Figure 11.4: Residuals of the calculation with the PIMPLE algorithm that is working in PISO mode. Figure 11.5: PISO mode. Velocit contours of the PIMPLE algorithm that is working in

130 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM PIMPLE working as PISO with large t Now we want to increase the time step to t = 0.025s, to reach the end time of the simulation much faster. This will increase the Courant number based on equation 11.3 and hence, the simulation should crash. Finall, we observe what we expect and after a few time steps the solver crashes with a Floating Point Exception division b ero. The residual plot demonstrates that after the evaluation of the flow, critical velocities are reached and the algorithm will break. 1e+00 1e-01 PISO algorithm p Ux U Residuals [-] 1e-02 1e-03 1e-04 1e Time [s] Figure 11.6: Residuals of the calculation with the PIMPLE algorithm that is working in PISO mode using a large time step. Figure 11.7: Velocit contours of the PIMPLE algorithm that is working in PISO mode with a large time step.

131 124 SIMPLE, PISO and PIMPLE algorithm In figure 11.7 the time step is given, when the instabilit is initiated. On the right top and bottom we can see areas where the velocit is alread larger than 60 m s total gra areas. Within the next two time steps, the instabilit moves further to the left side and the velocities increases to values larger than m s. Respectivel the pressure will blow up too. This happens until the accurac and the delineation of the computer results into a ero value PIMPLE algorithm modified add outer corrections Up to know, we run the PIMPLE algorithm in PISO mode. Now we will use the merged PISO-SIMPLE method b manipulating the algorithm. We can do this b adding the following kewords to the PIMPLE director. 1 PIMPLE 2 { 3 // Outer Loops Pressure - Momentum Correction 4 noutercorrectors 5; 5 } Listing 11.7: The control dictionar within the fvsolution file The noutercorrectors will set the ncorrpimple variable to five and hence, we make five outer corrections pressure-momentum correction loop. That include, re-building the velocit matrix with the new flux field, correct the pressure with the new velocit matrix and correct the fluxes based on the new pressure. Finall, we correct the velocities and go back to the re-guessing step till we reached five times. First of all, we would think that we will improve our calculation and make the algorithm more robust and stable but in fact the solver crashes. This can be seen in the residual and contour plot. In figure 11.9, the whole right part has alread incredible large velocities which will move on to the left, till, as before, the solver crashes. Furthermore, we see, that the crash happens alread earlier as before; compare residual plot The reason for that is the usage of the outer corrector loop. As we alread said, the PIMPLE algorithm is a combination of PISO and SIMPLE. Moreover, we know that the SIMPLE algorithm is not stable without relaxation. The outer corrector loops can be considered as a SIMPLE loop and thus without under-relaxation, the procedure can be unstable. Finall, it is not like the real SIMPLE algorithm because we have the limiting time step, that stabilies the whole solution procedure.

132 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM 125 1e+00 1e-01 PIMPLE algorithm p Ux U Residuals [-] 1e-02 1e-03 1e-04 1e Time [s] Figure 11.8: Residuals of the calculation with the PIMPLE algorithm and noutercorrectors = 5. Figure 11.9: Velocit contours of the PIMPLE algorithm with the usage of five noutercorrectors. The difference with noutercorrectors As we alread discussed, the difference with the usage of the outer loop correction is, that we recalculate the fluxes, pressure and momentum more often within one time step. All in all, we do this five times here. Doing a more detailed anale of the residuals, we get the plot shown in figure As we can see, within one time step, we have five more iteration steps SIMPLE. If we would put a line through the highest peaks of each quantit, we would get figure The reason for the crash can be explained as follow: If we go on in time, the flow field will further develop and within the small gap, we will get higher velocities.

133 126 SIMPLE, PISO and PIMPLE algorithm After some critical velocit is reached, the solution will diverge. The fact that we are looping fife times more over one time step will speed up the divergence after it is initiated and hence, the solver fails faster than in the case before. PIMPLE algorithm Time [s] e+00 p Ux U 1e-01 Residuals [-] 1e-02 1e-03 1e-04 1e Accumulated Outer Correction Loop [-] Figure 11.10: Residuals of the outer loop iterations PIMPLE algorithm further modified add inner corrections Commonl it is helpful and suggested, that we recalculate the pressure with the new updated fluxes but with the old matrix of the momentum PISO corrections - pressure correction loop. For that purpose we have to add a new keword to the PIMPLE dictionar as shown in the code snipped below. 1 PIMPLE 2 { 3 // Outer Loops Pressure - Momentum Correction 4 noutercorrectors 5; 5 6 // Inner Loops Pressure Correction 7 ncorrectors 2; 8 } Listing 11.8: The control dictionar within the fvsolution file

134 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM 127 As before, instead of getting a more stable algorithm, the solver alread crashes in the second time step. Therefore, we cannot anale the general residual plot but we can evaluate the curvatures of the outer and inner iterations. Recall: For one time step, we calculate five outer loops and within one outer loop, we calculate twice the pressure. Figure shows the inner and outer loops of the first two time steps. What we observe is, that within the first time step, the solution tends to diverge alread. This trend is continued within the second time step till the solver crashes. The result of the first time step calculation is given in figure Increasing the PIMPLE algorithm Accumulated Inner Correction Loops [-] e+00 p Ux 1e-01 U Residuals [-] 1e-02 1e-03 1e-04 1e Accumulated Outer Correction Loops [-] Figure 11.11: Residuals of the iterations of the inner and outer correction loops; noutercorrectors = 5 and ncorrectors = 2. Figure 11.12: Velocit contours; noutercorrectors = 5 and ncorrectors = 2.

135 128 SIMPLE, PISO and PIMPLE algorithm outer loops would lead to an even faster abort. This is because we can alread observe, that within the first time step, the solution tends to diverge. Using more outer iterations will help to establish the wrong solution and lead to a failure of the algorithm alread in the first time step. Up to now, it seems that the PIMPLE algorithm onl offers disadvantages. The reason for that is based on the wrong usage of the method. The first step for the correct usage of the algorithm is discussed in the next section PIMPLE algorithm with under-relaxation To make the PIMPLE algorithm work fine, stable and more robust, we have to think about the SIMPLE method outer correction loop. As we alread mentioned in section 11.1, the method is not consistent and hence, we are forced to use the under-relaxation technique; note that under-relaxation is not alwas necessar especiall for Co < 1 and non-stiff problems. Therefore, it is essential to tell OpenFOAM R the specific under-relaxation factors that should be used for the fields and/or equations. 1 PIMPLE 2 { 3 // Outer Loops Pressure - Momentum Correction 4 noutercorrectors 100; 5 6 // Inner Loops Pressure Correction 7 ncorrectors 2; 8 } 9 relaxationfactors 10 { 11 fields 12 { 13 p 0.4; 14 pfinal 0.4; // Last outer loop 15 } equations 18 { 19 U 0.6; 20 UFinal 0.6; // Last outer loop 21 } 22 } Listing 11.9: The relaxation dictionar within the fvsolution file

136 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM 129 The relaxation information is put into another dictionar within the fvsolution file. Note: The usage of the SIMPLEC algorithm can be activated b using the consistent keword. As we can see, we have two relaxation factors. One for all outer iterations except the last one and one that is used just for the last one. There is a big discussion about the value of the Final one. The question is, are we allowed to under-relax the final iteration like we did it for the previous outer loops? Actuall, if we under-relax the final outer loop we might loose some information and we are not consistent. However, if we have a lot of outer iterations alread done, it should be fine to use a value smaller than one for the Final iteration. This will stabilie the whole simulation because the relaxation factor of one in the last iteration can produce diverging results in some cases. To understand the cut of the information more, the different relaxation methods are discussed in more detail in section 12. If we do not specif an relaxation factor, the default value of one is used for all outer iterations. If we specif onl the the relaxation factor for p and U without the Final one, the final relaxation factors are set to one b default. Starting the simulation with the above mentioned relaxation factors and the increase of the outer correction loops to see what happens, lead to the results shown in figure and First of all, for each time step we are doing 100 1e+00 1e-01 PIMPLE algorithm p Ux U Residuals [-] 1e-02 1e-03 1e-04 1e Time [s] Figure 11.13: Residuals of the iterations of the inner and outer correction loops; noutercorrectors = 100 and ncorrectors = 2.

137 130 SIMPLE, PISO and PIMPLE algorithm Figure 11.14: Velocit contours; noutercorrectors = 100 and ncorrectors = 2. outer corrections and within one outer loop two inner corrections. That means, that we calculate for one time step, 100 momentum-pressure and 200 pressure correction calculations. Checking the time depended residuals, we get similar plots than using the PISO mode but now we have one essential difference. Within one time step we have much more iterations to find the correct solution; cf. figure And thats wh we are allowed to increase the time step without taking care that Co < 1. In the following case the Courant number is larger than 20 and the simulation is still stable. Comparing the contour plots of figure 11.3 and 11.14, we observe big differences. It seems that the PIMPLE algorithm smooths the whole solution. The reasons for that could be related to the too large time step that we were using within the PISO mode Co 1.3, the fact, that the Final under-relaxation factor was not set to one or that the time step within the PIMPLE mode was wa too large Co 20 and hence, we over jump some important and essential transient information which will influence the flow significantl. In our case, it is related to the last mentioned hpothesis. Based on the large time step, we do not recognie the establishment of a back flow vortex. Note: For high transient calculations, ou should keep in mind, that it is important to keep the time step in a range where all important phenomena that influence the flow field can be resolved. The current investigated case can be changed in a wa that the time step is adjusted b the highest Courant number. B using Co = 4 and re-run the simulation without changing an other setting, we get the result given in figure The new result is ver close to the PISO calculation compared to the result observed with a fixed time step.

138 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM PIMPLE algorithm speed up Instead of looping over the noutercorrectors till we reach the set value for each time step, it would be much better to leave the loop after we a defined residual limit is fulfilled for each quantit. For that purpose, we should also add the residual control sub dictionar to the PIMPLE dictionar. Doing that, OpenFOAM R will run the outer corrector loop until the residual criterion for each quantit is fulfilled. This will reduce the calculation time extremel and ensure good accurac within each time step. The next listing shows the correct usage of the PIMPLE algorithm. Of course, ou can also turn on or off the different switches we alread know. PIMPLE algorithm Residuals [-] e+00 p 1e-01 Ux U 1e-02 1e-03 1e-04 1e-05 Accumlation Inner Correction Loops [-] 1e-06 1e Outer Correction Loops [-] Figure 11.15: Residuals of the iterations of the inner and outer correction loops; noutercorrectors = 100 and ncorrectors = 2.

139 132 SIMPLE, PISO and PIMPLE algorithm Figure 11.16: Velocit contours; noutercorrectors = 100 and ncorrectors = 2; additional residual control added and Courant number controlled 1 PIMPLE 2 { 3 // Outer Loops Pressure - Momentum Correction 4 noutercorrectors 100; 5 6 // Inner Loops Pressure Correction 7 ncorrectors 2; 8 9 residualcontrol 10 { 11 p 12 { 13 reltol 0; // If this initial tolerance is reached, leave 16 tolerance 5e -5; 17 } U 20 { 21 reltol 0; // If this initial tolerance is reached, leave 24 tolerance 1e -4; 25 } 26 } 27 } relaxationfactors 30 { 31 fields

140 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM { 33 p 0.4; 34 pfinal 0.4; // Last outer loop 35 } 36 equations 37 { 38 U 0.6; 39 UFinal 0.6; // Last outer loop 40 } 41 } Listing 11.10: The residualcontrol dictionar within the fvsolution file The added residual control will speed up the whole PIMPLE procedure. We can now set-up 1000 outer correctors and the algorithm will automaticall leave the loop, after each residual criterion is fulfilled. The corresponding residual and contour plots are given below. 1e+00 1e-01 PIMPLE algorithm p Ux U Residuals [-] 1e-02 1e-03 1e-04 1e Time [s] Figure 11.17: Residuals of the iterations of the inner and outer correction loops; noutercorrectors = 100 and ncorrectors = 2; additional residual control added. The decrease of the calculation effort can also be seen in figure because within the same amount of outer loops, the solution is alread further with respect to the simulation time than in the case before here we have eleven time steps

141 134 SIMPLE, PISO and PIMPLE algorithm Figure 11.18: Velocit contours; noutercorrectors = 100 and ncorrectors = 2; additional residual control added PIMPLE algorithm Residuals [-] e+00 p 1e-01 Ux U 1e-02 1e-03 1e-04 1e-05 Accumlation Inner Correction Loops [-] 1e-06 1e Outer Correction Loops [-] Figure 11.19: Residuals of the iterations of the inner and outer correction loops; noutercorrectors = 100 and ncorrectors = 2; additional residual control added. whereas before we had onl seven within the same amount of outer loop iterations PIMPLE conclusion As we saw in the last sections, the PIMPLE algorithm can be used to enlarge the time step. It was presented and discussed how the algorithm should be used and which advantages it offers. For simple cases and flow pattern the PIMPLE method does not provide to much advantages. For more complex geometries, skewed, non-orthogonal meshes that include different kinds of cells and complex flow patterns or if we have to solve

142 11.4. THE CORRECT USAGE OF THE PIMPLE ALGORITHM 135 stiff sstems, the PIMPLE algorithm will provide much more advantages and can stabilie the simulation whereas the case would fail with PISO all the time or at least require extreme small time steps. In advance, the PIMPLE algorithm has to be applied correctl and the time step cannot be chosen as large as we want. We should alwas know which time scale can be achieved or which phenomena we are interested in. Different settings will lead to different solutions, if we do not take care about the numerics.

143 136 SIMPLE, PISO and PIMPLE algorithm

144 Chapter 12 The relaxation methods Relaxation methods are used for stead-state simulations or if one uses a large time step and the PIMPLE algorithm. The problem in such cases is that the solution might be not stable because the quantities change to much. E.g. we do not take care about numerical limits like the Courant number. To stabilie or even get a solution, we need to use relaxation methods. There are two different was to do that known as field relaxation and matrix relaxation Field relaxation The field relaxation is simple to understand and limits the new values of the field as follows: φ n+1 = φ n + αφ n+1, φ n n denotes the value of the last iteration, n+1, the actual calculated value, n+1 the new value and α is the relaxation factor. The implementation is given in the GeometricField class. If we use α = 1, the new value n+1 is identicall to the one we calculated n+1,. On the other hand, if we set the relaxation factor to ero, the new values will alwas be the old one. As we can see, the relaxation is limiting the change of the value of the quantit we are interested. Imagine a scenario where we calculate a quantit φ for each time step onl once while using a relaxation factor of α = 0.5. This would mean that we are not time consistent because we cut off almost 50% of the information. If we would make two corrections within one time step, then we would reach a better accurac but still, we would cut off information. That is wh OpenFOAM R offers the usage of 137

145 138 The relaxation methods the Final flag in order to change the relaxation factor for the final outer corrector within each time step. The field relaxation is mainl used for the pressure field Matrix relaxation In the field of computational fluid dnamics we are solving matrix sstems given as Ax = b. The matrix A is in general a sparse matrix. That means that we have a lot of ero entries. Solving this sstem in an iterative wa requires linear solvers. The performance of the solver is based on the diagonal dominance of the matrix A. Diagonal dominance means the following: If we would go through the matrix, each row has to be at least diagonal equal and one of the rows has to be diagonal dominant. What does that mean? Diagonal equalit is given if the diagonal element is equal to the sum of the magnitude of the off-diagonals, Diagonal dominant is given if the diagonal element is larger than the sum of the magnitude of the neighbor elements. The main difference between matrix and field relaxation is based on the fact, that we do not cut off too much information, if ever - it is based on the method how we relax. Relaxing the matrix means to make it more diagonal dominant and therefore the linear solvers are happier and will find the solution easier and faster. However, there are some things that we have to keep in mind. Relaxing the matrix means actuall to divide all diagonal elements b the relaxation factor α. Based on the fact that α is between 0 < α < 1, the diagonal elements increase its value. If we would onl change the values of the diagonal elements, the whole procedure is not consistent. To be consistent, we have to add the changes to the source vector b in order to keep consistenc. The whole procedure makes a better matrix sstem for the linear solvers but on the other hand makes it more explicit. The result is that we need more outer corrections to get all terms converged. Important is the pre-requirement for the matrix relaxation which introduce inconsistenc. The implementation is given in the fvmatrix class.

146 12.2. MATRIX RELAXATION 139 Pre-requirement for matrix relaxation The matrix relaxation requires the matrix to be at least diagonal equal in each row. Thats wh we have to manipulate the matrix before relaxing which introduces errors. To get an at least diagonal equal matrix A, we first check each row of the matrix and examine if we have an diagonal equal or dominant row. If the diagonal element is less than the sum of the magnitude of the off-diagonals, we replace the diagonal value with the sum of the off-diagonals. Here, it can happen that we loose information. However, the loose of information should me much less than for the field relaxation because the matrix A should have onl a few non-diagonal dominant entries if ever. But this is case depended see example below. Special matrix treatment with α = 1 Using a relaxation factor of α = 1 is different to the field relaxation. While the field relaxation would just do nothing, the matrix relaxation with might change the matrix; at least all rows that are not diagonal equal or dominant are manipulated to become diagonal equal. For example in cases where we have shock-waves, the cell in front of the shockwave will have a lower pressure value than the neighbor cell in which the shockwave alread exists. That will lead to non-diagonal dominant or equal entr in the matrix. To make the linear solver happier, we could make use of the matrix relaxation with the relaxation factor of α = 1. Thats wh for transonic cases the pressure equation is relaxed b α = 1; see rhopimplefoam cases with transonic behavior.

147 140 The relaxation methods

148 Chapter 13 OpenFOAM R tutorials For those who want to deal with OpenFOAM R and search for special tutorials, ou can checkout m website, that offers a lot of additional cases related to the following topics: Meshing with snapphexmesh, Solving and meshing different scenarios, Using the dnamic mesh librar, Setting up boundar conditions for AMI and ACMI, Generating own boundar conditions using the codedfixedvalue, Coupling DAKOTA R with OpenFOAM R, Coupling OpenFOAM R with Blender R. Furthermore, ou can find different libraries and publications an m website. For OpenFOAM R beginners If ou are not familiar with OpenFOAM R I recommend ou to check out the following website, wiki.openfoam.com. Here ou find a new wiki that offers a lot of information; ou can thank Jósef Nag for the good work and all contributers mentioned on the site. In addition ou will find some three weeks series in which ou will learn a lot of stuff. Furthermore, the User-Guide of OpenFOAM R should be read. 141

149 142 OpenFOAM R tutorials

150 Chapter 14 Appendix 14.1 The Incompressible Renolds-Stress-Equation The derivation of the Renolds-Stress tensor is structured as follows: The derivation of the time derivative is shown completel for all terms, The derivation of the convective derivative is shown completel for term a, The derivation of the shear-rate derivative is shown completel for term a, The derivation of the pressure term is shown completel for all terms. The derivation is given in all details to demonstrate how we get to the equation. Furthermore, we will be able to understand the terms and the reason wh we have to add terms in order to appl the product rule. Recall: In section 9.8 we introduced the wa how we will use the momentum equation in order to build the Renolds-Stress equation. Therefore, we multiplied the momentum equation with respect to the different fluctuations and set the sum of all terms to ero. Here, we introduced the Navier-Stokes operator N. Finall, we 143

151 144 Appendix build the equation that has to be evaluated which is given for completeness again: u xn ū x + u x + u N ū + u + u xn ū + u + u N ū x + u x a b + u xn ū + u + u N ū + u + u N ū x + u x + u N ū + u c d + u N ū + u + u N ū x + u x + u N ū + u + u N ū + u e f + u N ū x + u x + u xn ū + u }{{ + u } N ū + u + u xn ū x + u x g h + u N ū + u + u xn ū + u = 0. i Furthermore, we introduced the rules and tricks we are using during the derivation procedure: Renolds time-averaged terms that are linear in the fluctuation are ero, The derivative u i x i = 0, Product rule 1.2, Adding and subtracting terms to be able to use the product rule; gx = gx + fx fx. With that information, we will now demonstrate the derivation of the Renolds- Stress equation in the order given above. The Time Derivative a u xρ ū x + u x t + u ρ ū + u t = u xρ ū x t + u xρ u x t + u ρ ū t + u ρ u t = u xρ u x t + u ρ u t. 14.1

152 14.1. THE INCOMPRESSIBLE REYNOLDS-STRESS-EQUATION 145 b u xρ ū + u t + u ρ ū x + u x t = u xρ ū t + u xρ u t + u ρ ū x t + u ρ u x t = u xρ u t + u ρ u x t c u xρ ū + u t + u ρ ū + u t = u xρ ū t + u xρ u t + u ρ ū t + u ρ u t = u xρ u t + u ρ u t d u ρ ū x + u x t + u ρ ū + u t = u ρ ū x t + u ρ u x t + u ρ ū t + u ρ u t = u ρ u x t + u ρ u t e u ρ ū + u t + u ρ ū x + u x t = u ρ ū t + u ρ u t + u ρ ū x t + u ρ u x t = u ρ u t + u ρ u x t f u ρ ū + u t + u ρ ū + u t = u ρ ū t + u ρ u t + u ρ ū t + u ρ u t = u ρ u t + u ρ u t g u ρ ū x + u x t + u xρ ū + u t = u ρ ū x t + u ρ u x t + u xρ ū t + u xρ u t = u ρ u x t + u xρ u t. 14.7

153 146 Appendix h u ρ ū + u t + u xρ ū x + u x t = u ρ ū t + u ρ u t + u xρ ū x t + u xρ u x t = u ρ u t + u xρ u x t i u ρ ū + u t + u xρ ū + u t = u ρ ū t + u ρ u t + u xρ ū t + u xρ u t = u ρ u t + u xρ u t Sorting the terms, u xρ u x t + u xρ u x t + u ρ u t + u ρ u t + u ρ u t + u ρ u t +u xρ u t + u ρ u x t + u xρ u t + u ρ u x t + u ρ u t + u ρ u t +u ρ u x t + u xρ u t + u ρ u t + u ρ u t + u ρ u x t + u xρ u t and using the product rule, we end up with:. ρ u xu x t + ρ u u t + ρ u u t + ρ u xu t + ρ u xu t + ρ u u t + ρ u u x t + ρ u u t + ρ u u x t. The above expression can be written in one single term b using the Einsteins summation convention: ρ u j u i t = ρu j u i t

154 14.1. THE INCOMPRESSIBLE REYNOLDS-STRESS-EQUATION 147 The Convective Term First we will focus on the convective term of part a u x [ ρ ū x + u x ū x + u x + ρ ū + u ] ū x + u x x ū x + u x + ρ ū + u [ + u ρ ū x + u x x ū + u + ρ ū + u ū + u + ρ ū + u ] ū + u = u xρū x + u xρu x x ū x + u x + u xρū + u xρu ū x + u x + u ρū x + u ρu x x ū + u + u ρū + u ρu ū + u + u xρū + u xρu ū x + u x + u ρū + u ρu ū + u = u xρū x + u xρū x xūx x u x + u xρu x + u xρu xūx x x u x + u xρū ūx + u xρū + u xρu ūx + u xρu u x + u xρū ūx + u xρū u x + u xρu ūx + u xρu + u ρū x + u ρū x xū x u + u ρu x + u ρu xū x x u + u ρū ū + u ρū + u ρu ū + u ρu u + u ρū ū + u ρū u + u ρu ū + u ρu u x u x u u = u xρū x x u x + u xρu x + u xρu }{{ xūx x } x u x + u xρū u x + u xρu ūx + u xρu u x u xρū u x + u xρu ūx + u xρu u x + u ρū x x u + u ρu x + u ρu }{{ xū x } x u u ρū u 13 + u ρu ū 14 + u ρu u 15 + u ρū u 16 + u ρu ū 17 + u ρu u The same procedure can be done with the terms marked as b to i. Hence, we will end up alwas with the last line of equation result with respect to the

155 148 Appendix used quantities. Thus, we end up with: b u xρū x x u 19 + u xρū u 25 + u ρū u x 31 c u xρū x x u 37 + u xρū u 43 + u ρū u 49 d + u xρu x + u xρu }{{ xū x } 20 x u 21 + u xρū u 22 + u xρu ū 23 + u xρu ū + u xρu u + u ρū x x u x u ρu ūx + u ρu u x + u ρū u x + u ρu u xρu x + u xρu }{{ xū x } 38 + u xρu ū 44 + u ρu ū 50 x u 39 + u xρu u 45 + u ρu u 51 + u xρū u 40 + u ρū x x u 46 + u ρū u 52 + u xρu u 24 + u ρu x + u ρu }{{ xūx x } 29 ūx 35 + u xρu ū 41 x u x 30 + u ρu u x 36 + u xρu u 42 + u ρu x + u ρu }{{ xū x } 47 + u ρu ū 53 x u 48 + u ρu u 54 = u ρū x x u x + u ρu x + u ρu }{{ xūx x } x u x + u ρū u x + u ρu ūx + u ρu u x u ρū u x + u ρu ūx + u ρu u x + u ρū x x u + u ρu x + u ρu }{{ xū x } x u u ρū u 67 + u ρu ū 68 + u ρu u 69 + u ρū u 70 + u ρu ū 71 + u ρu u 72,,,

156 14.1. THE INCOMPRESSIBLE REYNOLDS-STRESS-EQUATION 149 e u ρū x x u 73 + u ρū u 79 + u ρū u x 85 f u ρū x x u 91 + u ρū u 97 + u ρū u 103 g + u ρu x + u ρu }{{ xū x } 74 x u 75 + u ρū u 76 + u ρu ū + u ρu u + u ρū x x u x u ρu ūx 86 + u ρu u x 87 + u ρu x + u ρu }{{ xū x } 92 + u ρu ū 98 + u ρu ū 104 x u 93 + u ρu u 99 + u ρu u u ρū u x 88 + u ρū u 94 + u ρū x x u u ρū u u ρu ū 77 + u ρu u 78 + u ρu x + u ρu }{{ xūx x } 83 + u ρu ūx 89 + u ρu ū 95 x u x 84 + u ρu u x 90 + u ρu u 96 + u ρu x + u ρu }{{ xū x } u ρu ū 107 x u u ρu u 108 = u ρū x x u x + u ρu x + u ρu }{{ xūx x } x u x + u ρū u x + u ρu ūx + u ρu u x u ρū u x + u ρu ūx + u ρu u x + u xρū x x u + u xρu x + u xρu }{{ xū x } x u u xρū u u xρu ū u xρu u u xρū u u xρu ū u xρu u 126,,,

157 150 Appendix h u ρū x x u u ρū u u xρū u x 139 i + u ρu x + u ρu }{{ xū x } 128 x u u ρū u u ρu ū + u ρu u + u xρū x x u x u xρu ūx u xρu u x u xρū u x u ρu ū u ρu u u xρu x + u xρu }{{ xūx x } u xρu ūx 143 x u x u xρu u x 144, u ρū x x u u ρū u u xρū u u ρu x + u ρu }{{ xū x } u ρu ū u xρu ū 158 x u u ρu u u xρu u u ρū u u xρū x x u u xρū u u ρu ū u ρu u u xρu x + u xρu }{{ xū x } u xρu ū 161 x u u xρu u 162. Analing the sum of terms, we figure out that there are different kind of terms: Terms that onl include the fluctuation quantities, Terms that include the mean quantit inside the derivation, Terms that include the mean quantit outside the derivation.

158 14.1. THE INCOMPRESSIBLE REYNOLDS-STRESS-EQUATION 151 Lets consider the terms that contains the fluctuation quantities for now: u xρu x x u x + u xρu u x + u xρu u x + u ρu x x u + u ρu u + u ρu u u xρu x x u + u xρu u + u xρu u + u ρu x x u x + u ρu u x + u ρu u x u xρu x x u 39 + u xρu u 42 + u ρu x x u x + u ρu 57 u x 60 + u xρu u 45 + u ρu u x 63 + u ρu x x u 48 + u ρu x x u 66 + u ρu u 51 + u ρu u 69 + u ρu x x u + u ρu u + u ρu u + u ρu x x u x + u ρu u ρu x x u + u ρu u + u ρu u + u ρu x x u + u ρu u x 87 u u ρu u 54 + u ρu u 72 + u ρu u x 90 + u ρu u u ρu x x u x + u ρu u x + u ρu u x + u xρu x x u + u xρu u + u xρu u u ρu x x u + u ρu u + u ρu u + u xρu x x u x + u xρu u x + u xρu u x u ρu x x u u ρu u u ρu u u xρu x x u u xρu u u xρu u 162 We get 54 terms that can be combined using the product rule. One example would be: ρ x u u u x = ρu u x x u + ρu u x x u + ρu u x u x In other words, three terms can be combined to one term. Appling the product rule to the terms, we will realie that not all terms can be combined. Therefore we.

159 152 Appendix have to add 27 terms of the following kind: ρu i u j u k x = k After adding these terms we end up with 81 terms that can be reduced to 27. Finall we get: ρ x u xu xu x + ρ u xu xu + ρ u xu xu + ρ x u xu u x + ρ u xu u + ρ u xu u + ρ x u xu u x + ρ u xu u + ρ u xu u + ρ x u u xu x + ρ u u xu + ρ u u xu + ρ x u u u x + ρ u u u + ρ u u u + ρ x u u u x + ρ u u u + ρ u u u + ρ x u u xu x + ρ u u xu + ρ u u xu + ρ x u u u x + ρ u u u + ρ u u u + ρ x u u u x + ρ u u u + ρ u u u. Now it is obvious that we can rewrite this equation using the Einsteins summation convention. In addition, we will put the densit inside the derivative due to the fact that it is constant. After appling the Renolds time-averaging, we end up with the convective term as: ρ u i x u j u k = ρu i k x u j u k k Now, we will consider all terms that contain the mean quantit inside the deriva-

160 14.1. THE INCOMPRESSIBLE REYNOLDS-STRESS-EQUATION 153 tive: u xρu x + u xρu }{{ xūx } ūx + u xρu ūx + u ρu x + u ρu }{{ xū } ū + u ρu ū u xρu x + u xρu }{{ xū } ū + u xρu ū + u ρu x + u ρu }{{ xūx } ūx + u ρu ūx u xρu x + u xρu }{{ xū } 38 ū 41 + u xρu ū 44 + u ρu x + u ρu }{{ xū } 47 ū 50 + u ρu ū 53 + u ρu x + u ρu }{{ xūx } ūx + u ρu ūx + u ρu x + u ρu }{{ xū } ū + u ρu ū u ρu x + u ρu }{{ xū } ū + u ρu ū + u ρu x + u ρu }{{ xūx } ūx + u ρu ūx u ρu x + u ρu }{{ xū } 92 ū 95 + u ρu ū 98 + u ρu x + u ρu }{{ xū } 101 ū u ρu ū u ρu x + u ρu }{{ xūx } ūx + u ρu ūx + u xρu x + u xρu }{{ xū } ū + u xρu ū u ρu x + u ρu }{{ xū } ū + u ρu ū + u xρu x + u xρu }{{ xūx } ūx + u xρu ūx u ρu x + u ρu }{{ xū } 146 ū u ρu ū u xρu x + u xρu }{{ xū } 155 ū u xρu ū 161 Again, we end up with 54 terms. This terms can be sorted and rearranged into twice 27 terms. Doing so, we will see that we can simplif the first 27 terms using the Einsteins summation convention to: ρu i u k ū j = ρu i x u k ū j k x k.

161 154 Appendix and the second 27 terms could be written as: ρu j u k ū i = ρu j x u k ū i k x k Note: If ou want to check if everthing is fine with the above equation, just build the sum of the two last equations and ou will see that ou get the 52 terms. Finall we have to consider all terms that contain the mean of the quantities outside of the derivative: u xρū x x u x + u xρū u x + u xρū u xρū x x u 19 + u xρū x x u 37 + u xρū u 22 + u xρū u 40 u x 7 + u xρū u 25 + u xρū u 43 + u ρū x x u x + u ρū u x + u ρū u ρū x x u 73 + u ρū x x u 91 + u ρū u 76 + u ρū u 94 u x 61 + u ρū u 79 + u ρū u 97 + u ρū x x u x + u ρū u x + u ρū u ρū x x u u ρū x x u u ρū u u ρū u 148 u x u ρū u u ρū u u ρū x x u 10 + u ρū x x u x 28 + u ρū x x u 46 + u ρū x x u 64 + u ρū x x u x 82 + u ρū x x u u xρū x x u u xρū x x u x u xρū x x u u ρū u 13 + u ρū u x 31 + u ρū u 49 + u ρū u 67 + u ρū u x 85 + u ρū u u xρū u u xρū u x u xρū u u ρū u 16 + u ρū u x 34 + u ρū u 52 + u ρū u 70 + u ρū u x 88 + u ρū u u xρū u u xρū u x u xρū u 160 To simplif these terms, we use the product rule 1.2 to combine two terms to.

162 14.1. THE INCOMPRESSIBLE REYNOLDS-STRESS-EQUATION 155 one. An example would be: ρū u xu = u ρū u x + u xρū u After combining the terms, we can rewrite the sum b using the Einsteins summation convention because the derivatives are alwas with respect to the mean quantities. Hence, we end up with 27 terms that can be expressed as: ρū k u i x u j = ū k ρu i k x u j k Now, the convective term is manipulated and derived. Combining all terms, we end up with: ū k ρu i u j x k + ρu ū i j u k + ρu ū j i x u k + ρu i k x k x u j u k k The Shear-Rate Term a { [ ūx u + u x µ x x x [ ūx + u µ x ū + u [ µ + ūx + u x x + ū + u x + ū + u ] ]} u ] [ ūx + u µ x + ū + u ] x { [ ū + u µ + ūx + ] u x x x [ ū + u µ ]} + ū + u = u x µ ūx u x x x x u x µ ū u x x u µ ū u x x x u µ ū u = u x x u x µ u x x µ u x u x x u x µ u x x u x µ ūx u x x x x x u x µ ūx u x x u µ ūx u x x u µ ū u µ u µ u µ u µ u x x µ u x u x u µ u x µ u u x u µ u x x u x µ ūx u x u x µ ū u x x u µ ū u u µ ū u µ u x µ u x µ u µ u x µ u u x u µ u x µ u u x u µ u x µ u x µ u µ u µ u x µ u

163 156 Appendix For the parts b to i we will just write the final terms: b u x x u x c u x x u x d u x u x e u x u x f µ u x µ u x x µ u x µ u x µ u x x µ u x µ u x µ u x x u x x u x u x x u x u x u x u x u x u µ u u x x x u µ u u x x x g u x u x x µ u x x µ u x u x u x x µ u x µ u x x µ u x µ u x µ u x x µ u x µ u x µ u x x µ u x µ u x µ u x x µ u x u x u u x u u u u u u u u u x µ u µ u x µ u µ u µ u x µ u µ u µ u x µ u µ u µ u x µ u u x u u x u u u u u u u u u x µ u µ u x µ u µ u µ u x µ u µ u µ u x µ u µ u µ u x µ u u x u u x u u u u u u u u u x µ u µ u x µ u µ u µ u x µ u µ u µ u x µ u µ u µ u x µ u u x u u x u u u u u u u u u x µ u µ u x µ u µ u µ u x µ u µ u µ u x µ u µ u µ u x µ u

164 14.1. THE INCOMPRESSIBLE REYNOLDS-STRESS-EQUATION 157 h u µ u u x x x u x µ u x u x x x x i u x u x x µ u x µ u x u x u x x µ u x µ u x x µ u x µ u x u u x u u x µ u µ u x µ u µ u u u x u u x µ u µ u x µ u µ u u u x u u x µ u µ u x µ u µ u u u x u u x µ u µ u x µ u µ u After the manipulation we find 102 terms. Again we want to put the fluctuation terms together, hence we need the product rule. Analing the sum, we can figure out that there is no wa to appl the product rule. The trick is simpl to add the missing 204 terms. One example is given now. Taking the term of f and of h, we get: u x µ u x u x µ u x This two terms cannot be merged, hence we need two terms in addition: 1 2 {}}{ u µ u µ u u x x x x + µ u u u }{{ x x µ u } x x added {}}{ µ u u x x + µ u u }{{ x x} added Now we are able to combine term 1 2 and 4 5 using the product rule. Furthermore term 3 and 6 are similar and can be combined too: x u µ u x 7 x u µ u x 8 +2µ u x u x

165 158 Appendix Now we see that the term 7 and 8 can be combined using the product rule again. Finall we end up with: µ u u + 2µ u u x x x x Repeating this procedure for all terms, we will reduce the alread existing 102 terms of a to i to 54. At the end we would realie that we can rewrite the sum of the 54 terms as: µ u i u j = µ u i u j x k x k x k x k The new introduced terms due to the trick, can be expressed as: Summing up, the shear-rate terms can be written as: 2µ u i x k u j x k = 2µ u i x k u j x k µ u i u j + 2µ u u i j x k x k x k x k The pressure derivative a b u p + p x + u x p + p u p + p x + u p + p x = u p x x + p u x x + u p + u = u p x + u x p + u p x + u p = p u x x + p u, p x = p u x + p u x,

166 14.1. THE INCOMPRESSIBLE REYNOLDS-STRESS-EQUATION 159 c d e u p + p x + u p + p u p + p + u x p + p u p + p + u p + p x f g h i u p + p + u p + p u p + p + u x p + p x u p + p + u p + p x x u p + p + u p + p x = u p x + u x p + u p + u = u p x + p u x + u p + u = u p + p u + u p x + u = u p + u = u p x + u = u p + u = u p + u p + p x + p + p + u p + u u x p + u x u x p x + u x u x p + u x p = p u x + p u, p = p u x + p u, p x = p u + p u x, p = p u + p u, p = p u x + p u x, p x = p u + p u x x, p = p u + p u x

167 160 Appendix Summing up and sorting: u p x x + p u x + p u x + p u x + p u + p u + p u x + p u + p u + u p x x + p u x + p u x + p u x + p u + p u + p u x + p u + p u. leads to the following expression: Finall we use the product rule, u i u i p + u p j x j x i p = p x u i + p u i, j x j x j u j p = p x u j + p u j, i x i x i p u i + p u i p x j x u j + p u j j x i x i to get the final form. B using the Kronecker delta function due to the fact that the pressure is onl in the main diagonal of a matrix, we get: [ ] u p i + u j + [ ] p x j x i x u j δ ik + p u i δ jk k Now the derivation of the Renolds-Stress equation is done. Putting all terms together, we can write the Renolds-stress equation in the following form: ρu j u i t + ū k ρu i u j u k + ρu ū i j u k + ρu ū j i x u k + ρu i k x k x u j u k k + 2µ u u [ i j u p x k x i + u j k x j x i µ u i u j x k x k ] + [ ] p x u j δ ik + p u i δ jk = k Appling the definition of the Renolds-Stress tensor 9.40, re-order the terms and multipl the whole equation b 1, we end up with the following form; recall: In

168 14.1. THE INCOMPRESSIBLE REYNOLDS-STRESS-EQUATION 161 almost all literatures we find the definition of the Renolds-Stress tensor denoted b τ. In addition we use the relation between the kinematic and dnamic viscosit: µ = νρ. Hence, we get: σ tji t + ū k σ tij u k = σ tjk ū i x k σ tik ū j x k + x k ν σ t ij + 2µ u u i j x k x k x }{{ k } ɛ ij [ ] p x u j δ ik + p u i δ jk k [ ] u p i + u j + ρu i x j x i x u j u k + k Π ij x k ρu i u j u k + p u j δ ik + p u i δ jk C ijk Finall, we can write the common Renolds-Stress equation: σ tji t + ū k σ tij u k = σ tjk ū i x k σ tik ū j x k + x k ν σ t ij x k + C ijk + ɛ ij Π ij

169 162 Appendix

170 Bibliograph J. Anderson. Computational Fluid Dnamics - The Basics With Applications R. Bron Bird, Warren E. Stewart, and Edwin N. Lightfoot. Transport Phenomena. John Wile & Sons, Madison, Wisconsin, June Jonathan A. Dantig and Michel Rappa. Solidification. EPFL Press, CH-1015 Lausanne, Switerland, first edition, J. Erickson. Hacking. dpunkt.verlag GmbH, Heidelberg, Berlin, first edition, Joel H. Feriger and Milovan Perić. Numerische Strömungsmechanik. Springer- Verlag, Heidelberg, Berlin, second edition, Christopher J. Greenshields. Open FOAM R Programmers Guide. OpenFOAM Foundation Ltd., version edition, December URL sourceforge.net/docs/guides-a4/programmersguide.pdf. Morton E. Gurtin, Eliot Fried, and Lallit Anand. The Mechanics and Thermodnamics of Continua Tobias Holmann. The pimple algorithm in openfoam, URL guide/the PIMPLE algorithm in OpenFOAM. Hrvoje Jasak. Error Analsis and Estimation for the Finite Volume Method with Applications to Fluid Flows. PhD thesis, Universit of London, Hrvoje Jasak and Henr G. Weller. Application of the finite volume method and unstructured meshes to linear elastics. In International journal for Numerical Methods in Engineering, August doi: /SICI :23.0.CO;2-Q. URL 163

171 164 BIBLIOGRAPHY Application of the Finite Volume Method and Unstructured Meshes to Linear Elasticit. Fadl Moukalled, Marwan Darwish, and Luca Mangani. The Finite Volume Method in Computational Fluid Dnamics An Advanced Introduction with Open- FOAMR and Matlab Fluid Mechanics and Its Applications, volume 113 of Fluid Mechanics and its applications. Springer-Verlag, doi: DOI / Rüdiger Schware. CFD Modellierung. Springer-Vieweg, Heidelberg, Berlin, first edition, H. K. Versteeg and W. Malalasekera. An Introduction to Computational Fluid Dnamics. Longman Group, first edition, David C. Wilcox. Turbulence Modeling for CFD. DCW Industries, 5354 Palm Drive, La Canada, California 91011, first edition, November 1994.

172 About the Author Hi everbod, m name is Tobias Holmann Bavaria, German and I work in the field of numerical simulations. The first time when I got in touch with OpenFOAM R was at the Fachhochschule Augsburg in I decided to write m Bachelor thesis using OpenFOAM R in the field of heat transfer. During m Master stud, I focused on the topics that are related to computational fluid dnamics and simulated different phenomena using ANSYS R CFX and OpenFOAM R. I finished the Master stud b writing a thesis about biomass combustion using the flamelet model. After that, I started to published m knowledge on m private website in order to help other people. Since 2014 I am a Ph.D. student at the Montanuniversitaet Leoben, investigating into local heat treatments for aluminum allos while coupling different phenomena and toolboxes in conjunction of developing new methods to predict the ield strength of the material. Since 2014 m colleague, Dr. Alexander Vakhrushev, taught me a lot of things in the field of numerical mathematics, matrix algebra, derivations and advanced programming in C++. During m spare time, I tr to figure out the limits of OpenFOAM R, give support in forums and develop new libraries and extensions for the toolbox in collaboration or alone.